The reason that you push with 6x BB instead of raising it to 3x BB is you gain more fold equity.
Fold equity is influence on your opponent to fold in simple terms. By pushing more chips into the pot, you are making your opponent risk a larger portion of his/her stack to call you.
So, if you push with 1000 chips left (blinds 100/200) and a player who has 2000 is thinking of calling, they would be risking half of their stack. If you only raised to 500/600, it would only be around a 1/4 of their stack to call.
You want to have as much fold equity as possible when you start to push with hands like Ax, Kx, all pocket pairs, and any two cards above 9 late in a tournament.