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I don't understand why these clauses led by what or that/which end with prepositions, why do we need a preposition at the end and what are the grammar rules.

What clauses:

We took an interest in what they are having a discussion about.

We don't understand what he is interested in.

That/which clauses:

The employee doesn't know the tasks which he is assigned to.

He drew a picture which students are afraid of on the wall.

This is the friend he can't stop talking about.

Are the sentences still correct if theose prepositions are omitted?

A simple sentence might be

He likes dogs. (SVO)

When made into a what question, the order changes:

What does he like? (OSV)

Now look at sentences that end in prepositional phrases:

He sat on a chair.

If we turn this into a what question there are two options.

On what did he sit? (move the preposition, formal question)

What did he sit on? (end in the preposition - informal)

What and which clauses have other uses besides asking questions. But in the example below, the word "which" has been moved to the front of the clause.

The chair on which he sat

The chair which he sat on

In both cases, the pronoun "which" refers to the chair, which is an indirect object of the verb "sat", and would be used with a preposition after the verb: "sat on the chair" But if the order is changed to OSV, you can move the preposition or leave it. It is optional.

The proposition is required:

The chair which he sat

Would make "chair" the direct object of sat. And "sat" doesn't have a direct object (in this sense)

In your examples, the prepositions are required. Omitting them would change the meaning or make the sentence ungrammatical.

Without the prepositions, the sentences are either incomplete or ambiguous, or potentially have different meanings.

It's worth noting the strict old-fashioned grammarians say you shouldn't end a clause with a preposition. Following those rules often sounds excessively formal, though sometimes something sounds 'wrong' and sounds better reorganised like that. So, some of your later examples would, according to those rules, become:

The employee doesn't know the tasks to which he is assigned.

This is the friend about whom he can't stop talking.

I say this in case you're dealing with someone who believes in those rules. Dealing with most native speakers, you can generally ignore it.

So, looking at each of your examples, you want to know if you can lose the preposition. Let's see.

We took an interest in what they are having a discussion.

That is incomplete and doesn't make sense. If you're interested in the discussion, rather than the subject of it, you could say:

We took an interest in their discussion.

If you are interested in the subject, it's:

We took an interest in the subject of their discussion.

(You could choose other words than subject)

We don't understand what he is interested.

Again, incomplete and nonsensical. You could rearrange, but you still need to indicate the object of his interest, not the fact of his interest. Or you could rephrase:

We don't understand where his interests lie.

It's putting it a different way, but it has the same meaning.

The employee doesn't know the tasks which he is assigned.

That changes the meaning, flipping object and subject, but it ends up meaning the same thing. The version in your question means the employee is assigned to the tasks, and the version immediately above means the tasks are assigned to the employee.

He drew a picture which students are afraid on the wall.

That's fairly meaningless. There are two possible missing words; either "picture in which" or "afraid of on the wall" - and they mean different things.

This is the friend he can't stop talking.

That looks like there's a missing comma between friend and he, which would mean that 'this' is 'the friend' (presumably previously mentioned), and that said friend can't stop talking.

Pronouns, prepositions, conjunctions - they all matter. Sometimes they can be implicit, but don't assume they can be unless you have reason to be sure.

We don't need a preposition at the end, but we do need one somewhere in order to connect an expression of time, space etc. to some noun, verb or adjective etc. The omission of a preposition altogether would render the sentence ungrammatical, or at least change the meaning.

The alternative to ending the sentence with a preposition is to 'front' it, by placing it at the beginning of the relative clause along with its complement. Take you first example:

[1] We took an interest in what they are having a discussion about.

[2] We took an interest in the thing(s) about which they are having a
discussion.

Here, they were having a discussion about something and we took an interest in whatever it was that they were discussing. [1] with its stranded preposition "about" is the perfectly normal way of conveying this fact. But [2], which is designed to avoid preposition stranding, is ridiculously formal, and best avoided here.

The same applies to your other examples.