I need to see who favorite my tweet. I need twitter api call to know who favorite my tweet and the person details.
There isn't a Twitter API call at present to find out who favorited a tweet. You could attempt to pull the information out of one of the live stream APIs as discussed here and similar to this FAQ, although the data is real time and not historical. Favstar already does this, so you could have a look at that (it has a rudimentary API that may help).
Bear in mind that you won't see favorites of users who protect their tweets with any of these methods though.