Question: Is the phrase "I'm glad it helped" grammatically correct?

And if it is, does it express correctly that I am more than happy that I could help someone?

Answer: 
"I'm glad it helped" is grammatically correct. But it doesn't mean that you're glad *you* helped, it means you're glad that something helped. It might, from context, be clear that it was something you said, did, or gave them. But "I'm glad it helped" alone doesn't say that.

You can say "I'm glad I helped" or "I'm glad I was able to help" to indicate what you want. But you can't say "I'm glad it helped" unless it's clear what it is you are referring to -- some specific thing that helped.
This is very much comprehensible and commonly used, but it is not grammatically correct as the word "that" has been omitted. I know that it's pedantic but don't tell the guy that his sentence is absolutely correct. Of course, I would definitely advise a foreigner in England to use this phrase as it is polite and idiomatic.