You're offering to take someone somewhere for a coffee and you'll pay. "I'll treat you" is the general expression for offering someone a "treat" or a little pleasure of some sort.

No thanks. I don't drink coffee, but when you "treat" someone to coffee, or dinner, or a trip to the cinema, it means you're paying.

It is 'treat you to', not 'treat you for' also.

In British English, it is more common to say 'Can I shout you a coffee?'