You use "it turns out that ___" when you got some information wrong, and you're now giving the correct information. You first have to say what you thought was true. (Or the listener might already know this from other conversations you've had.) Then you say something like:

It turns out that it's on Wednesday, not tomorrow.

So it turns out that I had been waiting on the wrong side of the train station.

You can use this phrase in the past tense too:

I thought it was at ten. It turned out that it was at eleven.