Air is invisible because it is a gas, and gas particles generally allow light to pass through them because they are so spread apart. Gases expand to fill any container they are in, as they do not hold a shape; therefore, it is usually easy to see through them.
One way to think of the way gas works is to think of putting stickers on a window. If a kid just put two or three stickers up, which represent atoms in a gas, it is still easy to see around them. However, if the kid puts up more stickers, such as with the atoms in a liquid, it becomes more difficult. If nearly the whole window is covered up in stickers, like the atoms in a solid, it becomes nearly impossible to see, though there are exceptions to the rule in all cases.