“Pushing” is the act of killing enemy minion waves to the point that your minion waves can start attacking the enemy’s walls, towers and fortifications. This is important to work on constantly because:
1) It eats up tower ammunition –
Towers only have a certain amount of ammunition and they stop shooting when they run out. The more you push, the quicker they run out and the quicker they die.
2) You can get early “free” damage against towers –
The tutorial tells you about how towers and forts hit minions before hitting you. This means that you should be moving in (if enemy heroes aren’t nearby) to hit their towers whenever possible. Did only 3 of the minions in your wave make it to their defenses? Fine, run in, get 1-2 attacks off on the towers then fall back. Do this over and over again and the towers will fall. You don’t need to kill towers all at once, take them down piece by piece.
3) Destroying a fort hurts them and helps you –
Each fort that is up spawns a wave of minions. Destroy the fort and you stop those minions from spawning. This leads to being able to push that lane easier and being able to apply more pressure. Forts also grant considerable xp for your team both in the early game and the late game. Destroying a simple wall can also greatly reduces the amount of defense they can run behind when they are in trouble.
All in all effective pushing is important in Heroes, especially early on.