It was an emergency vehicle. On its way to an emergency. They don't like to stop or slow, because it could mean someone's life. In a normal context, they would have been blasting through intersections with sirens howling and lights flashing, and trusting other drivers to get out of their way.

But an airliner can't dodge easily, the laws of physics are brutal, and a (likely tired and overworked) air traffic controller, the guy who was supposed to keep this from happening, screwed up.

How do you not see a plane coming in to land?

It's. An. Airport. In an area dense with airports and air traffic, at that. The driver of an emergency vehicle in supposed to be looking in front of them, not above them. The airplane would have been above, to the side, and some distance away, and the driver would probably have assumed it was further away than it was or on a slightly different vector, if they noticed it at all.