Your mission, if you choose to accept it.. is to assume the role of Dig Dug (later named Taizo Hori years after the game's release) and defeat monsters while digging underground.
You have two methods with which you can kill your enemy.
One option is to blow them up with your air pump until they explode. The other option is to dig underneath a rock so that it falls on the enemy, crushing them.
Another strategy is to blow up the monster partially with your air pump as it stuns them for a few seconds, allowing you to pass by safely.