It means the same thing as
I asked my friend what happened.
The form "I asked of my friend what happened" is grammatically correct, but it is simply archaic, and as such not often used anymore. The easier option, without preposition, is also the one you will see most often used, so (for once?) the easy option is the best one.
Some extra background, just informational, about the specific sentence:
Because it is not commonly used, it may actually cause confusion, because people may think you mean to say:
I asked what happened to my friend.
I actually read the sentence in you question initially like that, "correcting" the preposition in my head, because I wrongly assumed you used the wrong preposition.
So you might give people the feeling you mean something else than what you say, even though what you say is grammatically correct. Since communication aims at getting the right message across, I think it is better to use I asked my friend to avoid sounding archaic, and possibly, confusion.