Meaning of: "'If factoring large integers is hard, then breaking RSA is hard,' is unproven"
The easiest way to think about it is to think of the contrapositive.

The statement: 


  if factoring large integers is hard, then breaking RSA is hard


is equivalent to the following:


  if breaking RSA is easy, then factoring large integers is easy


This statement has not been proven.

What they're saying is, assume we have an algorithm that solves factoring in polynomial time. Then we can use it to construct an algorithm that solves RSA in polynomial time.

But, there could be some other way to crack RSA that doesn't involve factoring integers. It's possible that we will find we can crack RSA in a way that doesn't let us factor integers in polynomial time.

In short, we know that RSA is at least as easy as factoring. There are two possible outcomes: RSA and factoring are of equivalent difficulty, or RSA is a strictly easier problem than factoring. We don't know which is the case.