There have been a number of replies on this post, and it seems to me most of them have missed the boat. The phenomenon of the lion killing the cubs of a 'free' female he encounters is common, but by no means universal in the animal kingdom. It's been observed in chimps, too, but (to my knowledge) that's all.
As to why the lion does it - it doesn't have a reason. It does it because it seems the natural thing to do (to him). Just as (first example that jumps to mind) stalking things upwind of him, rather than downwind of him, seems the natural thing to do. And both behaviours would have developed the same way...some sort of mutation.
Imagine the hypothetical first generation of lions. They don't actually care about cubs, one way or the other. But in one lion, something in his genetic makeup mutates, and to him the natural course of things is to kill all cubs he sees. This doesn't help, as he kills his own cubs, as well...and his particular mutation dies out. But another lion has a slightly different mutation...he thinks it seems natural to kill all cubs he sees except those of lionesses with whom he has recently mated. So he ends up killing all the cubs he comes across who couldn't possibly be his. Coincidentally (as far as he's concerned - it certainly isn't why he does it) this works pretty well as a survival strategy for his genes, since he doesn't let any cubs survive except those that might be his own, and the lionesses he mates with don't 'waste' their time (from his point of view) caring for cubs that can't be his. The result of this is that he produces more offspring than other lions, who don't care at all about cubs, one way or the other. His offspring tend to inherit his mutation, and think killing other lions' cubs is natural, too...and so the trait spreads, until lions with this mutation become the norm...at which time (to be pedantic) it stops being a mutation.
But don't think the lion does it to bring the female into heat, or to stop her wasting her time on other lions' cubs...he just does it because it seems natural to him. An analogy I can think of is catching something thrown at you. If someone throws a ball at your face, you catch it, or knock it away. You don't do the complex differential calculations required to work out the object's parabolic arc in your head; you just instinctively knock it away because that seems the natural thing to do. The lion doesn't weigh up the benefits of killing off the cubs...he just does it, because it seems the natural thing to do.
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