The situation is fairly complex so I'm not surprised it was confusing. Here's the general rundown (partially pulled from this article for brevity)

1. Sometime in July, Ford (Kavanaugh's accuser) wrote a letter to Diane Feinstein (D-CA) with her allegation that Kavanaugh had assaulted her sometime around 1982. The letter purportedly requested anonymity and Feinstein appears to have honored that request.


2. On September 12, The Intercept was fed the letter by an unnamed source. At this point, the confirmation process was nearly complete and the Judiciary Committee was set to hold a vote.


3. The next few days saw rapid development of the story, as well as two less credible accusations. Ford's name was exposed and reporters interviewed people named in the letter. Democrats would start to demand hearings during this time.


4. On September 27, both Ford and Kavanaugh testified in front of cameras and the Judiciary Committee.


5. On September 28, the Judiciary Committe voted, entirely along party lines, to move the nomination to the Senate Floor. Senator Jeff Flake (R-AZ) asked the FBI to investigate the allegations before a final vote.


6. On October 4 the FBI reported back. The gist of the report (not made public) is that none of any of the allegations of Ford or the other two accusers made could be substantiated. The report was criticized for many reasons, most notably that the FBI had not interviewed Ford.


7. On October 6, Kavanaugh was confirmed 50-48

How did this just get fast tracked? Where is the politics in the middle?

It wasn't fast tracked. The process was nearly done (almost 2 months of interviews, investigations and meetings with Kavanaugh) when the allegations leaked. The problem for Kavanaugh opponents generally came down to

1. The timing. There's no denying the leak was of tremendous benefit to Democrats. Prior speculation was that red-state Democrats would vote to confirm. The allegation ensured that didn't happen (with all except Manchin citing this as the reason they voted "No"). But it also meant that timing before the mid-term elections were seen as stalling for a post-election vote, when Democrats could potentially retake the Senate and block the nomination outright. There had already been attempts to stall the voting prior to this, and it was widely viewed among Republicans that this was an extension of those attempts. With Republicans in control, there was no convincing them to entertain any more delays.


2. A lack of credible allegations. Ford was the only one of the accusers who made any allegations that could be taken seriously (the last accuser changed her story during a TV interview). She testified, with conviction, that it was Kavanaugh who had assaulted her, but nobody she named could corroborate her story, or even when it could have occurred. The FBI appears to have been unable to shed any light on when it could have happened. The other accusations had even worse credibility, with the second admitting she was quite drunk, and the third making outlandish accusations (that Kavanaugh was part of a sex ring).


3. Ultimately, the vote on confirming Kavanaugh was about whether or not Senators, not the FBI or police, believed he was guilty of this crime. Having had additional (if criticized) background checking done was unlikely to sway too many people beyond what they had already decided. The politics were already set before the allegations.

I'm not sure what the standard for evidence is (i.e. for references) on this site.

If you want to know, here's what I inferred from following the Twitters of a couple of (anti-Trump) American lawyers.

• The weren't "sexual assault hearings", they were "senate confirmation hearings"


• Senate procedure has previously (since 2013 and 2017, as explained in detail in another answer) been changed to require only a bare majority, rather than 60%


• News of a historic sexual assault complaint was leaked somehow, late in the process (i.e. just before a vote was due)


• They delayed the vote (for about a week) to hear from the accuser in person


• After those hearings, one Republican said, let's have just a little bit more of that FBI background check


• There's no criminal complaint involved (and if anything the complaint would be of a state crime not a federal crime), so the FBI's investigation is a background investigation (requested by the White House), and not an independent criminal investigation -- the FBI has no mandate to investigate except to whatever extent the White House asks them to


• The White House asked for a limited investigation -- e.g. the FBI didn't interview the accuser, nor the accused, nor review other potential evidence (e.g. therapy notes), nor (so far as I know) did they fact-check what Kavanaugh said during the hearings, nor did they accept testimony from other people (possibly including other independent complaints) who were trying to volunteer it -- which took little time (i.e. completed within a week)


• The results of the investigation were kept secret (not published)


• The Senate majority (IOW the Republican party) then went ahead (within a few days) and voted, having concluded e.g. that the allegations were unproven and the accusation uncertain (to put it politely)

Some complain the scope of the investigation was purposefully limited, to result in a shallow sham, a fig leaf to provide deniability ("we investigated and found nothing").

Conversely, the President said (yesterday's speech at the White House) that Kavanaugh was "proven innocent".

Essentially, the Republicans had decided they were going to confirm Kavanaugh and do it fast in order for him to be on the bench for an October case. At that point, the hearings lasted only long enough to get enough people on board.

Supreme Court justices are, by the Constitution, nominated by the President and confirmed or not by the Senate. There are no other Constitutional requirements, which means the Senate can conduct whatever hearings it likes before voting. There is no requirement for any serious deliberation.

Here is a "Tick Tock"-style article about the end of the nomination and the vote. (This link is to the NY Times, but it's an AP article and available elsewhere, too.) It covers what happened when, but it boils down to:

1. Committee votes to pass Kavanaugh, with the condition that the FBI do an investigation.


2. White House authorizes a one-week investigation. (There are accusations that the FBI was hobbled in this investigation.)


3. Four Senators (Republicans Collins, Murkowski and Flake, Democrat Manchin) were undeclared during that week. Because of the 48 committed Republican votes for, and 48 committed Democratic votes against, it required two of these four to agree to pass Kavanaugh in order for him to be confirmed (VP Pence breaks ties).


4. The FBI released its report to Senators on Thursday.


5. All Senators reviewed it.


6. Collins, Flake, and Manchin announced they would vote for Kavanaugh after all.


7. The cloture vote happened on Friday, and passed 51-49.


8. The confirmation vote happened on Saturday, and passed 50-48 (Murkowski voted "present" instead of "no", to balance the lack of another "yes" Senator who couldn't make the vote).


9. Kavanaugh was sworn in in a private ceremony on Saturday.

It's not rocket science. It boils down to three things:

1. Republicans wanted a win going into the midterms to boost their appeal


2. Republicans wanted to confirm Kavanaugh before the midterms, because if they lost control of the Senate they would never be able to confirm him (or anyone like him)


3. They had the votes, and so could get away with it.

History will tell us what Manchin, Flake and especially Collins were promised if they voted to confirm. Probably soon.

Weakening of the Filibuster was one of the primary reasons this moved faster than previously possible. This weakening of the filibuster began in 2013, by then Senate Majority Leader Harry Reid (D-Nev.). Reid justified his action by saying: "The American people believe Congress is broken. The American people believe the Senate is broken. And I believe they are right."

Under provisions of the Senate Rules, a small minority of senators has, in the past, been able to hold up nominations despite the majority of senators wanting the nomination to proceed. One of those obstructionist tactics, the filibuster, takes advantage of the fact that Senate Rules do not limit the duration of debating a topic under consideration on the senate floor. Overcoming a filibuster had traditionally required a 3/5 (60%) supermajority vote in a process called cloture, and changing the rules a 2/3 (66%) supermajority vote.

However, in November 2013, Senate Democrats used the "nuclear option" to overcome a threatened filibuster by Republicans on executive branch nominations and federal judicial appointments. This permitted President Obama's nominations for federal judges to be moved forward more rapidly that traditionally possible. In turn, in April 2017, Senate Republicans themselves used the nuclear option to advance the nomination of Neil Gorsuch to a floor vote over the objections of a minority of democratic senators.

Thus by the time Judge Cavanaugh's nomination was being debated on the Senate floor, use of the "nuclear option" to defeat filibuster techniques on nominations was somewhat unsurprising. The Senate majority leader used this tactic to limit debate on the senate floor to 30 hours, and after the alloted time for debate, that a vote was held. The motion to confirm Judge Kavanaugh was passed by a narrow margin, though it was still a majority (but not supermajority) that supported it.

Indifference

Even after Republicans learned of the allegations a short time before the public did, they still planed to confirm Kavanaugh anyway without investigating.

It wasn't until after the press and public learned about it, and Kavanaugh's angry, rant-filled, and often dishonest interview about it, that a few Republicans requested a limited "investigation" (the others still planned to go ahead and confirm him).

Sure the investigation wasn't thorough by any means, but they they can at-least say they did investigate and found nothing. That gives them some political cover.

Politics

It's important to remember that current Republican president Donald Trump has had many more accusations of accusations of sexual misconduct and assault. What are they going to do? Start taking these sexual assault charges seriously while their sitting president has even worse allegations against him? Unlikely, as we saw with recently failed Senate nominee Roy Moore.