You use it and if the Pokémon kills you before you make another move, the Pokémon who deleivers the finishing blow dies too. Kind of like an explosion for weak Pokémon in the attacking stat like Mismagius.
Let's give you an example...
My Mismagius is at pretty good HP and the enemy has their Donphan out. I know I cannot KO with Shadow Ball and Assurance will do a ton, possibly fainting me depending on how much attack his Donphan has (standard spinner Donphan is Adamant with 252 attack EVs).
So, seeing as I have no safe switches, I use Destiny Bond as he uses Assurance. It takes me down to about 10% HP
Now the opponent, being a total n00b goes for an Ice Shard and kills Mismagius, thinking he will avoid Destiny Bond... WRONG!!! As long as you don't make a move (that includes ALL moves, btw... status moves, attacking moves etc...) after you use Destiny Bond and the opponent kills you, they die too =)
Note that DB doesn't work if you are statused in the sense that if you use DB, they use a move that brings you to ~5% HP and then you die that turn due to Toxic damage, they are NOT taken down with you because the opponent didn't deliver the finishing move. Even if they are the one who used toxic on you, it still won't work. The same goes for weather damage and things like leech seed damage.
Hope this helped =D