Square integrable function

So Ive got a module on quantum mechanics but theres something I dont quite get. Ive put an arrow next to it. What is it saying in simple terms haha?

Any explanation is appreciated.

It just means that integral is finite, over whatever region you are integrating over.

So its saying if I had limits that when i integrated it you would be able to get an exact value out?
Whats the issue if my integral is infinite?

Yep, you get some finite value.

Dunno what the issue is when it's infinite, I haven't covered this theory. But it seems to me that this statement is just an agreement, i.e. an axiom of sorts, so you just have to take it at face value and deduce your theory based on this fact. Maybe later on you will find out the necessity for it being finite.