I would say that the molecules themselves (that is, all the molecules of O₂, N₂, CO₂, etc. which make up our atmosphere) probably don’t change density very much, on average, with respect to temperature. At least not in the range of temperatures we generally experience on Earth. Why do people say that hot air is “less dense” than cold air, then?

This has to do with the density of the air as a whole, rather than the density of the individual molecules. A gas like air is basically a jumble of bunch molecules spaced out with nothing in between them. It's like if you took a piece of paper and randomly placed 20 dots on it. Some of the dots would end up close together, but most of them would be pretty spaced out, with no other dots nearby.

As temperature increases the space in between these molecules increases. Why is this? Let’s imagine we have a balloon full of air which is completely sealed off so no air can escape or enter the balloon. If we could zoom in really close to a part of the air in the balloon so that we could see all the individual molecules, they wouldn’t be sitting still, they would be flying all around and bumping into each other. Suppose we uniformly increased the temperature of the air inside the balloon (and that the balloon is a perfect insulator so none of the heat escapes). This causes air molecules in the balloon to travel further and faster than they were before we raised the temperature. This is because we are transferring energy to the molecules in the balloon by increasing the temperature. When they collide with the inside walls of the balloon they begin to push it outwards because they have more energy than they did before (they are "pushing harder" against the walls, in a sense). This causes the balloon to expand (i.e. volume to increase) - even though the number of air particles in the balloon hasn’t changed. This means the density of the air inside the balloon decreases.

So, in summary, a higher temperature means more space between individual molecules, which means that the volume of the air has to get bigger, which means that the density has to go down.