Why is it so windy in Las Cruces, NM?

The most likely reason I can see is essentially a combination of two factors. One, there is a developing surface low over far southeast Colorado which is causing the surface pressure gradient over the southern high plains to become increasingly steep, and enhancing the wind. Second, the mid level wind flow is stronger than usual as the storm track is pretty far south. Warming of the lower layers of the atmosphere may be allowing for some of this faster wind to be brought down to the surface.

It's been a long time since I've been to Las Cruces, but as I recall there is not a lot vegetation in the surrounding area to slow down the winds, that's probably the biggest contributing factor. Also, the proximity to nearby mountain ranges can cause a mountain-valley circulation and acceleration of winds through narrow mountain valleys.