Veiling glare is light that's not intended to be part of the image, per se, but ends up on the recording medium (film or sensor) anyway. It's caused by reflections and scattering of light by optical elements and the lens barrel.
This produces an overlay of general brightness, which raises what should be the darkest parts, reducing overall image contrast.
For example, imagine shooting on a sunny day, and framing a photograph which doesn't include the sun directly, but where direct sunlight still falls on the front glass of that lens. Some of that light still makes it to the sensor, even though it doesn't represent the form of anything in your image.
Smears or dust on the lens (or on an attached filter) can scatter light in unintended ways, compounding the problem.
Adding additional lens elements — like a glass filter, either for protection or for a special effect — can make this worse, for several reasons. First, it's another piece of glass, and usually a flat one at that. Second, many filters are of low quality and don't have good coatings. And, since they're usually right on the front of the lens, further away from protection from out-of-the-image light, they're prone to making the problem worse even when of high quality.
Your biggest defense is a lens hood — or otherwise keeping the front element of the lens shaded. All light that strikes the front element has the opportunity to scatter and bounce around in the lens, causing veiling glare — and bright sunlight can easily wash out the image.
Lens bodies are designed with matte black internal finishes and often have baffles and other features to control reflection. And, lens surfaces are given special coatings in part to minimize this reflection.
The visible lens flare — sometimes called "ghosting" seen in photographs or movies are related, but not quite the same. In those cases, the light is more focused and controlled, causing a bright highlight shaped like the aperture, or sometimes rays or lines. These are can also be caused by having bright lights hit the front element — like, say, having the sun in the frame! — but veiling glare can be there even if you don't see any of that.