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 certainly grammatical, and it expresses the meaning you indicate. There are variations including, for example, *I’m glad I was able to help*. More briefly, you can say *My pleasure*.
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.