What does a camera’s low-pass (anti-aliasing) filter do, and is it better to have one?

A low-pass (or anti-aliasing) filter adds a little bit of blur to the image, which is helpful in avoiding moiré patterns. Moiré patterns are caused when two similar patterns interfere with each other, causing a much more noticeable (and usually unwanted) pattern, like this (image from P. Fraundorf):

What you have here is two sets of parallel lines that, when placed slightly askew and one atop the other, cause a pattern of dark and light lines. The pixels in a digital camera's image sensor are arranged in a grid, and any sort of closely-spaced vertical or horizontal lines in your image can therefore create a moiré pattern in the final image. The low-pass filter adds a small amount of blur that doesn't significantly change the image overall, but still reduces the moiré effect.

It's true that some cameras come without the low-pass filter. A great example is the Canon 5DS R, which is exactly the same as the 5DS except that it lacks the low-pass filter. And you can see from the product page that Canon warns about moiré with this camera:

The possibility of moiré and color artifacts is greater due to the LPF cancellation function.