Why is a circle not simply-connected?

Imagine you have a rubber band and you lay it down on top of a circle drawn on a piece of paper. You're asked to stretch/shrink the rubber band until it's crumpled up at a single point; however, the rubber band must stay on top of the circle at all times, and you can't cut it. Intuitively this does not seem possible. This corresponds to the topological fact that $\pi_1(S^1)=\mathbb{Z}$. This means there are loops in $S^1$ (the circle) which cannot be continuously shrunk to a point. Hence the circle is not simply connected.

Now if you're allowed to move the rubber band on top of the circle but also inside the circle (i.e., in a disk), it's easy to crumple it up to one point - just push every point on the rubber band in a straight line toward the point where you want it to end up. No matter how the rubber band is initially arranged within the disk, you can still crumple it to a point. This corresponds to the topological fact that $\pi_1(D^2)=0$. That is, every loop in $D^2$ (the disk) can be continuously shrunk to a point (via a straight-line homotopy as described with the rubber band). Hence the disk is simply connected.

A circle is the perimeter without the interior. Intuitively, it isn't simply connected because there are two ways around without the interior. To shrink the circle to a point you have to break it. The disk is simply connected. Any closed curve can be shrunk to a point continuously.