"it speaks to me" means that the phrase has great meaning to you personally. The concept resonates with you, due to personal experience, etc.

When you find yourself at one of those conceptual art exhibition but you're not sure what to make of said art, and you don't want to look like and idiot in front of the artist, you just say, mmm, it speaks to me.

Hi webkadoris! If something like a proverb speaks to you, it means something like: you like it, you understand it at a deeper level, you agree with what it's saying. I hope that helps.

It has simply a special meaning or importance for you.