Is there any benefit of stretching out the kill shot?
While in many cases it probably is just showing off and it is generally considered bad sportsmanship, playing cards and killing enemy minions is worth experience.
Additionally, Ravers points out that they may be trying to finish a quest, such as "Destroy 40 minions", "Play 40 spells", "Play 20 minions that cost 5 or more", or "Play 30 minions that cost 2 or less".
What determines how much experience I get from a hearthstone match in play mode?
How is this behaviour called?
The general term for the behaviour that you are describing is BM (short for "bad manners"). It is generally considered bad sportmanship, as already mentioned, but if you watch competitive Hearthstone you sometimes see players BM each other. The reason that this is OK, is because they know each other (maybe are even close friends) and take it as a joke. It is also quite entertaining for the viewers.
What are the reasons for it?
On the ladder players may do it to show dominance, often accompanied by several Sorry-Emotes, but the most common reason is finishing daily quests.
Maybe the xp-gain mentioned in the other answer plays a role too, but from my experience the it is all about finishing quests, like "Destroy 40 minions" or "Play 20 minions that cost 5 or most".
How to handle?
Since Blizzard made the change that conceding the game is not part of the action-queue anymore, you can just concede to skip all the BM your opponent may throw at you.
Before this change you would have to watch all your opponents actions if he queued them up before you pressed concede. Since they fixed this it is instant and saves you all that stuff. So best advice is to concede if you opponent has obvious lethal on the board.
However, there are situations where you dont know if you are dead and your opponent does a lot of other stuff before he kills you with his topdecked fireball. In this case you cant do anything but stay calm and queue up for the next game :)
Am I the only one who doesn't consider such behaviour bad manners? It's simply part of the game. Sometimes you want to be quick, sometimes you want to be absolutely sure and sometimes you want to feel the power rush coming with finally beating a tough opponent. Pick one or mix and mingle, there's plenty of reasons to stretch-out a game.
The major point to keep in mind is you can always get out by conceding. The other player isn't wasting your time. He's playing the game the way he wants to within the scope of the rules. If you disagree with his playing style there's nothing stopping you from conceding and moving on.