We are squeezing the air in the syringe together when we push on the plunger, but we can only squeeze the air together so much, and then the plunger won't go any farther, because the air molecules are squeezed as close together as we can squeeze them by pushing on the plunger.

When you push the plunger of a syringe without your finger plugging the hole, the force that you are applying doesn't have an "opposing force", or a force pushing back. This allows whatever is in the syringe to come out of the hole because of the pressure applied to the fluid inside. When you cover the exit, the fluid that pushes on your finger is blocked and an opposing force is applied to the fluid. This makes it much harder to push the fluid back since the two forces cancel out and the net movement is 0.

It is because there is air inside of the syringe, and you are not letting it escape. Air exerts a pressure on anything it touches, and the denser the air, the greater the pressure. By compressing the air inside of the syringe, you are making it denser, to the point where you are no longer strong enough to keep pushing the plunger in. The force of the air on the plunger equals the force of your finger pressing on it.

When you put your finger over the nozzle of a syringe, you are making a closed volume of air. When you push on the plunger, you are decreasing the volume of air inside the syringe by increasing the pressure. More pressure makes the closed volume of air even smaller. In order to press all the way down, you need to decrease the volume to zero. This can only be done if you can apply infinite pressure to the plunger.

Unfortunately, it does not matter how much you work out your thumb because you won't be able to keep covering the nozzle when the pressure becomes extreme. Your finger will hurt! So, what happens if we put a perfect seal on the nozzle and use a very strong machine to push on the plunger? Will we get to infinite pressure? Unfortunately, the answer is "No!" The syringe will explode before you can get to infinite pressure.