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In the following sentence, it seems it should end up with "like it" or "like there". Why the both of "it" and "there" have come together?

I have been to Washington several times, and I like it there.

In the sentence, it does not refer to Washington. So, you shouldn't be interpreting the sentence in this way:

✘ I've been to Washington several times, and I like (Washington) there.

Instead, it is a dummy pronoun. From Wikipedia:

A dummy pronoun, also called an expletive pronoun or pleonastic pronoun, is a pronoun used to fulfill the syntactical requirements without providing explicit meaning.

In the sentence, it does have some meaning, but it's not exactly defined. For a native speaker, the meaning of the sentence is generally the same as:

✔ I've been to Washington several times, and I like (something that exists) there.

A conversation could go something like this:

"I've been to Washington, and I like it there."

"Oh, really? What's there that you like?"

The use if it is non-specific. It refers to something, but the sentence doesn't clarify what it is. However, it does say that it exists there in Washington.

There is also a contrast between here and there (and anywhere else):

I hate (the weather) here in Antarctica, but I like (it) there in Washington.

The use of (it) has changed from a dummy pronoun to a referential pronoun because the first part of the sentence has actually made it clear what (it) is that's being referenced. But the important part is the contrast between here and there.

Compare this with the sentence in which there has been omitted:

I've been to Washington several times, and I like it.

In this sentence, no dummy pronoun is used. It is a referential pronoun that refers to Washington.

Although awkward, the sentence could be rephrased in the following way:

I've been to Washington several times, and I like Washington.

A similar conversation could be constructed around this:

"I've been to Washington, and I like it."

"Oh, really? What do you like about Washington?"

Note the specific difference between this conversation and the earlier one.

Whereas it in the first conversation refers to something completely undefined (aside from its existence in Washington), it in this conversation refers to something about Washington itself.

There are, I think, two ways to look at this:

1. There is a concept or idea, something like [x] makes me happy or I am content with [x], which started off being spelled / pronounced I like [it], and then got moved into a (grammatically) different situation where "I like it" is really "I am happy", so if you read it simply as though it says …and I am happy there you'll see the it as vestigial (like an appendix) and not really part of the meaning.




2. The it can be like the it in it is raining, i.e. a sort of global, catch-all reference to the milieu of the speaker, i.e. I moved to [place] and conditions there are great for me.

Personally I suspect both are true.

Following is your answer which I found on web.
"The use of the pronoun 'it' is called an 'empty' object/subject.
We use it as a meaningless subject with expressions that refer to time, weather, temperature, distances, or just the current situation.
In the sentence "I like it here", 'it' refers to the situation or the conditions.
For example:
It's ten o'clock.
It's Monday again."

This is the same "it" of "It's raining."; and "It's hot today". You could probably say, "I have been to W., and I like it.", "it" being Washington. But "... I like it there." sounds a little more common to me, and "it", in this case, isn't Washington itself in a concrete sense. I'm pretty sure it's the "it" that describes the environment/ambience/existence in general.