There are two primary models of the universe. One is the expanding model, which says that the universe is constantly increasing in size. The other is, the steady state model, which says that the universe is approximately constant in size and shape. What evidence would you expect to see if the expanding model were correct?

Ok, suppose that the universe is expanding:

If the universe is expanding, then the distance between "fixed" points in the universe (we can set the fixed points as planets, galaxies, etc) would change with the time.

This means that, if the distance between the Earth and a given star or planet (that is not in our nearby region) should change as time goes by.

Now, this was actually observed:

There is something called the doppler effect:

In a short way, if an object is moving away, the light that comes from that object will have a red shift (it seems a little more reddish than it actually is)

If the object is moving towards us, the light that comes from the object will have a blue shift (it seems a little more blueish than it actually is)

Those shifts can be observed from the Earth, so from that is safe to conclude that the expanding model is actually correct.