To simply understand it instead of the 3D visualization, isn't it just a cylindrical pipe, whose outer surface is the same as the inner surface? And endless one at that because it continues forever without hard edges, or in other words, don't have an inside. For example, if you do turn it into a puffy cylinder to smooth the edge, it becomes clearer to you how it's now a 3D structure, than 2D.

For the 2D surface to have the inside same as the outside, it must be joined together in a way that isn't possible in 3D, but is possible in 4D.

The simplest way to imagine this as walking forward on a straight road until you walk the same path in the same direction, but through the opposite side -- would lead to you imagining a path looping with a half-twist, and then it makes intuitive sense why the Mobius strip has a half-twist.

It is not a miracle that happened by twisting the strip, but rather, the idea of a non-orientable path is only possible with the half-twist.

This also explains gives you insight into why the Klein bottle can be formed by attaching two Mobius strips (or looping one in 360 degrees) - because it's the same way a cylinder can be built from a plain strip.

Applications

In QFT, Klein bottles can be used to represent 3-dimensional structures called "qubits".
Someone's idea on LinkedIn, which is a nice way to think about it to look for a theory: Spacetime Quarks as Klein Bottles | James Cupps | LinkedIn Pulse

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