Why is the clause "They filled me in on all the latest news from Cambridge" correct?

To fill in is very similar to “to fill” – the “in” is an adverb that gives the nuance of putting something into an empty space that should not be there so that there is no longer a blank or a hole or an empty space.

For example, you can fill in a hole in the road by putting stones into it; you can fill in an official form or a question sheet by putting writing into the empty answer boxes.

”To fill in”, in the sense you ask about, is an extension of this idea. They put in (= add/insert) news about Cambridge that you lacked and so the fill in the missing part of your knowledge:

“They filled me in on all the latest news from Cambridge” – They added news so as to fill in the news that I was missing.

The verb fill when used in the sense inform has a specific complementation pattern:

fill + object + in + (prepositional phrase headed by on)

The object is the entity being informed and the noun phrase contained in the prepositional phrase headed by on is the content of what that entity is being informed of. The prepositional phrase headed by on is optional.

The complementation pattern you suggest wouldn't work even with the common definition of fill:

*They filled water in the bottle.

Prepositional verbs often have different complementation patterns than the verbs they are "based" on and the order of elements is fairly rigid.