That's a really good question. There are two main forces acting on a plane in flight that control how fast it can go. These are called thrust and drag. Thrust is the force that the airplane uses to move forward through the air, and drag is the force that the air exerts on the plane to slow it down.

Thrust in an airplane comes from its propellers or jet engines. These act in the same way as a propeller does on a boat. If you place a solid sheet in a fluid (both water and air are fluids, just water is much more dense) and move it back and forth you will see it push the water. The force that pushes the water backwards in return pushes the sheet forward. Whenever a force pushes on an object, it pushes back. The equal and opposite force is what you feel if you push against a wall. The reason that you do not fall over is that the wall pushes back against you.

In the case of a propeller there are several small sheets of metal that rotate very quickly. Since the propellers are at an angle, as they spin around they push the air in front of them towards the back of the plane, and in return the air pushes back and moves the plane forward. A jet engine essentially has a series of small blades rotating at a very high speed inside the engine.

Drag is a force that all planes feel. This is a form of friction. Friction is the force that pushes back on a textbook as you push it across the table. Planes can be designed to minimize their drag by building them in very streamlined shapes. Imagine you are in a swimming pool and you push yourself off of the wall. If you put your arms straight out to your sides you are not going to go very far before the friction of the water against your body (this is drag) slows you down. However, if you put your arms straight at your side you will go farther because there is less drag in this streamlined case. By having a plane with engines that produce a large amount of thrust, and designing it to be streamlined, an airplane can travel very quickly.