The wiki states that the lucky foot raises luck. Unfortunately, it fails to state just what that means. So, what does raising my luck do?
In addition to the other answer, I have experimentally found that the lucky foot increases the chances of winning at the shell game in the arcade. The wiki says that the probability of winning on any single run is 33%, but when playing as Cain I have recorded a greater than 50% win rate (so far it seems to be 66%).
Edit: according to discussion on the item page on the wiki, the foot also makes you only get good pills. The comment said that collecting it turned a bad pill into a good pill like the PhD item.
Edit 2: it appears that the Womb levels decrease or nullify the effect of the Lucky Foot. I went to an arcade on The Womb 2 and found that my win rate was below 50%.
I tested with 140 consecutive runs of the shell game in The Womb 2 and I won 77 times, so it looks like the win chance is a bit over 50%, but definitely not 66%.
In addition, the lucky foot increases the chance of getting an item from the beggar. Without the lucky foot, it often takes 4 or 5 coins before the beggar will drop a single item. With the lucky foot, however, I have never seen it take more than 3 coins for him to drop something.
If you have a look at the items page on the wiki, there are a couple of people there who state the foot does several thing:
While as a programmer it makes sense to me that all the above are based on some global "luck" factor, which the lucky foot improves, I can't say I have actually noticed any of the above in one of my playthroughs (that doesn't mean that the effects aren't too subtle to easily notice).
What I did notice, personally, is that when I play as Cain, who starts with the lucky foot, I tend to get more pick-up drops once I clear a room of baddies.