Sure man! Disclaimer: I am no expert. Most of this stuff is just gathered from reading/speaking to folks with experience so there will certainly be pieces of this that can be better explained by others...
The basic premise is that you don't place nearly as much of a logistical burden on the enemy when you kill as opposed to injure. For example, injuries require immediate battle medical attention, transportation, long term care, etc etc. Death doesn't.
I've read a decent amount on the subject and you're not likely to find any "official" sources stating the above (you can imagine why) so some of it is just based on word of mouth and theory I'm sure re: the injury deal. I will also say that I'm not 100% sure it serves the soldier well (compared to .308/7.62x51) but hell what do I know in that regard...
The 5.56 also fragments better than 7.62x51 so it could, certainly, do more long term damage but we've also come a long way in ammo development since we switched so who knows how that's affected the whole deal.
The 5.56 has significantly less recoil so that was surely a factor. Means more rounds placed down range. So combine the ability to carry more with shooting more and you've got more bullets towards the enemy which increases the probability of hitting the target.
In other words, if a frontline soldier can get 4 5.56 rounds out for every 3 7.62's (being conservative here, that number is probably higher) you've just upped your probability by a large margin.