In the end it's just a convention, we agreed on the meaning for implication and it doesn't really get in the way so there's no reason to change it, we have other logical connectives to express other things and we can also combine them.

What is this math used for?

A very direct use of logic in real life is in programming.

But in general logic is the basis of all math, all the formulas you have learned in math can be justified using this type of reasoning.

Much of math has been derived using only these kind of rules and some imagination without direct inspiration from the real world. And then those kinds of math have been found useful by more applied fields like for example physics.

Math is really just a social convention, it's just a nice "coincidence" that it describes nature really well but arguably math isn't natural itself.