What does "I'll go you one better" mean?

I'll go you one better, seem as you were so polite to ask me for permission.

Can someone please explain this to me? I've never heard the "go you one better" part.

Does this mean that a person will do something better than asked for?

This is a friendly colloquial expression which is used in two ways (you have correctly stated one of them in your question).

A willingness to provide something even more than requested

— Please could you tell me the way to the city centre?
— I'll go you one better, and drive you there.

— Do you know in which year the Berlin wall fell?
— I'll go you one better: I know the exact date and time.

A response which is similar to but greater than a previous statement by another person

— My computer was running very slowly today.
— I'll go you one better, mine stopped working entirely.

A variation I've heard is: I'll do you one better. Same meaning as in: I'll do even more than you ask me to/ for.

It means "I will outdo you" or "I will go a step further than you did". It is not necessarily either favourable or unfavourable.