The shape of the bokeh is related to the apparent shape of the aperture of the lens.

Straight on, this will produce a bokeh that is approximately a circle. As the subject moves away from the center of the field, the bokeh starts to look like a sliver of the circle.

This can be reduced by stopping down the lens.

More on this can be read at Shape of the blur patch and Cat's eye effect.

To achieve this affect, you need a wider field of view in the lens (a long lens will never see the aperture from a steep angle) and shoot wide open.

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The optics of this can be understood by looking at the light rays through the lens system:

However, its not quite this simple as the lens construction itself also plays into the shape of the bokeh.

The only difference between the four images is the lens. Each lens has a different front and rear pupil size and lens blade count and shape. If this effect wasn't caused by the lens, then it would indicate some other phenomena. But instead, we do see a difference between different lenses on the same camera indicating that the lens construction is the cause for the shape and the place one should look to understand the nature of this bokeh.

Ultimately, the cat's eye bokeh is a form of mechanical vignetting similar to when the lens hood is too long for the lens and blocks some of the scene.

This type of bokeh can also be seen in the apparent changing of the shape in DIY bokeh shape filters:

Note the shape of the heart and mickey mouse at the edge of the frame. If one was to go and look at the lens from those point light sources, one would see that the shape of the heart or mickey mouse on the camera is the shape of the bokeh (with some additional adjustments for the apparent shape of the pupil).