In this demonstration, students discover that variances in air pressure can be exploited to create suction.
When the demonstration balloon is small, it fits in side the cup, it is curved and takes up a lot of room in the cup. As the balloon inflates, less of the balloon is in the cup, increasing the volume available to the air molecules trapped inside the cup. This reduces the air pressure inside the cup since the air molecules now have more space to move around. Meanwhile, the higher pressure air outside the cup pushes the cup into the balloon.
In other words, the "suction" is really the pressure of the air outside the cup pushing the cup into the balloon and causing it to stick there.
A great visual is to introduce the idea of a "push of war" (instead of a "tug of war"). The (higher) pressure of the air outside the cup pushes the cup into the balloon harder than the (lower) pressure inside the cup pushes out.