Where can I find information about MO activity which led to progress in somebody's research? E.g. is there an efficient way to cite MO contribution in a paper in such a way that it will be easily detected here?
There is a thread on meta dedicated mainly to tracking MO's impact on (published) research, see Best of MathOverflow, or papers inspired by MathOverflow which continued http://mathoverflow.tqft.net/discussion/64/where-to-keep-track-of-mathoverflow-success-stories/ (on the old meta board).
I do not know of a way (other than posting it there) that actually ensures that it will be noticed here. However, using the recommended form for citing, which one should do anyway, will presumably increase the chances it is picked up whenever somebody searches for mention of MO in the literature.
To get this recommended form (also in bibtex and amsrefs format), just click 'share' below a post and then 'cite'.
You might also be interested in Improving citations of MathOverflow posts that discusses ways to improve the ways in which MO is or can be cited.