Biological death clock
There are biological clocks you can implement in order to enable this to happen. Let's say, for instance, that these creatures have a unique brain which can only function in the presence of a rare metal to serve as a cofactor, say, zinc. However, zinc is not normally found in these creature's diet, so the way these creatures get zinc is very odd. The way I would do is make these creatures amphibious - they are born in the water and consume zinc-containing life forms there to finish developing to a land creature.
Now, these creatures will slowly use up zinc over the course of a lifetime and have no way of replenishing it under normal conditions. In order to keep their zinc from suffering from Le Chatelier's principle (that things tend towards an equilibrium), they have a special organ whose's sole job it is to keep surplus zinc. The organ fills up on zinc during the aquatic stage of life and can only hold so much.
But, should a creature find and devour another creature's zinc organ, then their own body can process that zinc and send it to it's own zinc organ. It'd get the remaining years of zinc, though obviously not all the zinc perfectly. At that point, it'd have added a few more years to that death clock, but it won't stop them from biological decay, so even with this, they'd still die from brain deterioration.
It should also be noted that once these creatures get to the level of modern medicine they can just make supplements and skip this whole issue. It should also be noted that I'm not really sure why the creatures would evolve to have all these very specific conditions.