So let's talk about chirality first so that definition is covered. Your left and right shoes are chiral mirror images of one another, since they are clearly like one another, but there's no way to rotate a right shoe to turn it into a left shoe and vice versa. Another example, this time of a 2D chiral object, would be a spiral. A spiral spins either clockwise or counter clockwise, and no rotation in a 2D space can change that. You need to rotate the spiral in a 3rd dimension to get it to become its mirror image. You might do the same to a shoe, but you'd have to rotate it in a 4th dimension since it's a 3D object.

So a good test of orientabilIty is this: take a lesser-dimensioned chiral shape and traverse it along the shape of choice. If there exists no traversal which can make the chiral object look like its mirror image, then the shape is orientable. This can also be said as the shape having clockwise and anti-clockwise as distinct directions. Both the Möbius strip and the Klein bottle are non-orientable because they can convert lesser-dimensional chiral objects into their mirror images simply by traversing those objects along their surface in the right fashion.

You can imagine tracing a path along a Klein bottle to see that it only has one side. To get more precise than that requires some topological context. If you slice it down the middle it turns into two Möbius strips and an orientation of the Klein bottle would induce an orientation of the strips, which are non-orientable. Alternately it has zero top integer homology, which you can get from looking at a triangulation. The orientable double cover of a Klein bottle is a torus, which is connected (if it were orientable, the double cover would be two disconnected Klein bottles).