What I mean by lens speed is how much light it allows through to the sensor or put in another way, it’s wide open aperture. The larger the aperture (the smaller the f number), the more light is passed through it and hence the faster the exposure.
There are a number of advantages to having a large aperture lens, with a wider the aperture the depth of field is shallower. Which allows you to isolate your subject from the foreground and background by bringing them out of focus.
With wider aperture you can take pictures in a darker environment without needing a tripod or a boost in ISO or a slower shutter speed which could cause a blurry image due to hand shake.
As a general rule DSLR lenses have a bigger wide aperture than point and shoot lenses but this depends on the lens of course. The more expensive pro lenses tend to have wider apertures than do consumer lenses. Having a larger aperture means the lens needs to be built larger also and needs larger lens elements which drive up the cost.