Why do players always use smite as the killing blow?

Solution 1:

SaintVicious, jungler for the CLG.NA team, once said it on stream. If you are certain that you are not going to be counterjungled, then sure, smite right away. However, circumstances can quickly change while you are attacking the creep (assuming smite is used on objectives like buffs).

Situation #1: You are jungling but leave base a little late. Like usual you head to Red and you feel confident so tell your team you don't need their help leashing it. So you smite right away and begin autoattacking it. In the animation of your last auto-attack, the enemy jungler smites it away from you over the wall, then turns and kills you with the help of the small lizards and the red buff they now have cuz your smite was down.

Situation #2: You have been jungling and cleared the jungle already. As typical, you get ready to feed your AP Carry the blue buff. While, waiting for the AP Carry to deal the finishing blow, the whole enemy team comes charging from river to the Blue led by their jungler. Fortunately this time, you saved your smite, so before the enemy jungler can smite steal it, you secure blue for your team. While not optimal, certainly better than the enemy team having both blue buffs.

Finally, it is good practice as it helps you use smite to secure important objectives like dragon and baron. After all practice makes perfect, and if you say that you always use smite first except for dragon and baron, you won't be as practiced. A few seconds lost on the smite cooldown is hardly noticeable most of the time, but gaining a little more practice on using your smite to secure kills never hurts anyone.

Solution 2:

I imagine it is mostly done to prevent the enemy from showing up and finishing off the target.

If you wait until a smite will kill it, you have much less chance of the enemy AP mid coming over and sniping it with a skill shot or the enemy jungler showing up to steal it with their smite.