その大学しか受からなかったからそこに行くことにしたって英語でなんて言うの？

All three of these answers are good for explaining to someone that you chose the college/university that you are at because it was the only one that you were accepted into.
When you 'gain entry' into a school/college/university it means that you have either had an application accepted or you passed the entrance examination.

A simple way of phrasing the sentence is to say: This is the only University that I have chosen to attend.

A) using "thus" will mean that your first clause ("This was the only school I was accepted to") is the reason for your second clause ([why] "I decided to go there").
B) By stating "The reason why I decided to go to school here" at the beginning of the sentence you are being more direct with what you are ultimately trying to say. The second clause of the sentence explains why.

I decided to come to this university because they were the only one that gave me an offer.

Here a decision was made because there was no other option.
The word "they" is used because the university has more than one person in it.

I am attending this university because they were the only ones that gave me a place to study here.

In this sentence you may be confused by the word "ONES" because we are speaking about more than one university having the ability to give an offer.

If you are allowed to do something then they say you are 'accepted' to do it
If you make a decision on what to do then you say you 'decided' to do that
'only' means just one (only one) you could also say 'no other'

Well it seems like a sad story but you can always make a joke about it as in the example sentence.
If you have a limited choice, it means that you have few options in a certain situation.

When we talk about getting in to college we often talk about 'admission' or 'being admitted' so this is a standard word to use. 'Chose' means 'decided on'.

Decision is based on - This means why you made that decision.