図書館で本を借りたって英語でなんて言うの？

By using the term 'to borrow' or 'borrowing' means that you are not keeping the item but just going to use and return it from where you got it from.
So if you are borrowing books from the library then you could say 'Im borrowing/borrowed books from the library'

To borrow is to to take something with the intention of returning it at a later date, so you have 'borrowed' your books from the library.
The term 'checking out' refers to the action of making the library staff aware of which books you have taken as you leave. So you could also say that you have 'Checked out some books.
To loan something is to give it to someone with the intention that they retutrn it. The opposite of borrowing. So we can say that we have something 'on loan' which means that we borrowed it.

When you mention using or having library books, there is never any need to use the word 'borrowed'. The function of a library is to to lend books to the public, so both lending and borrowing are functions automatically accepted when you speak about libraries.

The two sentences you see provided above are excellent ways to express to your listener that you borrowed some books from the library. In the second sentence you will notice the phrase check out. Although this term has different meanings, in this sentence it means to borrow something from someone with the intent of returning out. This phrase would make a great addition to your vocabulary.

You can use any of these sentences :-
1. I borrowed some books from the library:-
This statement uses the word "borrow" which means to take something for a period of time, expecting to return it at an agreed upon time.

1. I took the book on the loan from the library :-
   To take something out on loan is another way of saying that it was "borrowed". The expectation is that the book will be returned to the library at a later date.