How can I visualise the fundamental group of the projective plane?

View $\mathbb{RP}^2$ as the Möbius strip with a disk attached on its boundary circle. The core circle of the Möbius strip is the generator of the fundamental group. You cannot drag the core circle to its boundary - it's "stuck". To see this, try and perturb the core circle.

I've included a picture which shows how wrapping around twice gets you "unstuck".