If you try an experiment with a regular balloon, you might be able to picture what happens more easily. Take the balloon and inflate it just a little, and stop. Draw two dots on the balloon with a pen, then inflate it again. You can think of the two points as points on the ground above the magma chamber. With very sensitive instruments, geologists can detect very very small movements of the earth's surface, and see if the magma chamber below is inflating or deflating.
Geologists can also measure the tilting of the ground above the magma chamber.
A final way that scientists have evidence of magma chambers comes from the chemistry of the lavas that are erupted. Geochemists think the have a pretty good idea of the composition of magma that is generated in the Earth's mantle. However, the compositon of the lava that gets erupted on to the surface often is different. It may have too much olivine, or not enough olivine, or other more subtle differences. It is often very easy to explain these differences by processes that go on in magma chambers.