My teacher poured water over some soil and said that the air bubbles prove that soil contains air. However I was wondering if there is a more direct approach to proving that soil contains air, like explicitly separating one from the other. Is there an experiment like this? Also, how would the air look like? I've never exactly "seen" air, so how does one know when the air has been separated? Maybe by making it a liquid (liquid nitrogen)?
Your teacher's method of pouring water over soil is an excellent way to demonstrate that there is air in the soil. If you're still unconvinced, it's possible to directly separate the air out. Here's how:
You fill a vacuum chamber with your soil. All of it. So it's all soil and no free space is left. Now you just pump the air to a different chamber. There are several ways to do it so I will not elaborate on the mechanics. This method extracts the air from the intergranular porosity (i.e. the space between the soil grains) to your other container.
Also, how would the air look like?
Well, it doesn't look like anything, because it is after all air. It is possible to liquefy it by cooling or compressing. It's also possible to measure the composition of the gas you extract out of it and see that it matches the composition of air. It will not be exact, depending on the soil. Some soils have decomposing organic matter (dead bacteria, dead plants, dead other things) so they might have slightly less oxygen and slightly more carbon dioxide, methane, sulfur dioxide, hydrogen sulfide, and a bunch of other things that usually also smell.