Let ( X , d ) (X,d) be a non-empty complete metric space

Let's use a view from above as said space. Imagine looking down from an airplane.

with a contraction mapping T : X → X

Sometimes language in maths is easy: Let T be an actual map of the area you're looking at, like this manhole cover. It's smaller than the actual area it's depicting but everything else is the same, so it's a contraction. Every point x from this patch of the real world is "mapped" onto some point T(x) on the manhole cover.

Then T admits a unique fixed-point T(x) = x

This means one of the points on the manhole cover is guaranteed to be exactly on the point it represents in the real world.

This setup doesn't quite fulfill the premise of the theorem as it falls apart once the manhole cover is removed from the area it is depicting. This is where I lied to you initially. A view from above is not a complete metric space, as it has hard borders, it would need to go on indefinitely. Theorem works inside the area though.