As many have said, the most likely explanation is a serious case of adrenaline. As for how he managed to get a foothold, it also is a part of adrenaline. In addition to allowing for the possibility to tap into normally impossible reserves of muscle usage, adrenaline also partially numbs the body to pain. Where only seconds before his leap, Simba was having real trouble getting a foothold (due to the amount of pain it would put him through to dig his claws in far enough to secure his position), after the adrenaline kicked in, he was able to suck it up and dig his claws in for a split second so that he could use his front and back legs to propel himself up the cliff.
The real question is why (aside from the obvious need to further the plot) Mufasa wasn't able to pull off a similar feat when Scar was about to throw him into the Wildebeest stampede?
My only thought is that perhaps he had already used so much energy to sprint to the scene, jump into the stampede, save Simba, and then pull himself 90% of the way up the hill, that his body simply didn't have anything more to give, adrenaline or not.