Are you talking about the in-cabin air filter, or the engine air filter?
In either case, the answer is no. I mean you can do it, but the filter will not work like it is supposed to afterwards. The reason for this is they are made of paper. When you get them wet like you are suggesting, it destroys the element.
As to the why you have to change them, once they become completely full of dirt, they restrict air flow and can cause issues in engine performance. It can also push dirt through, negating even having an air filter in the first place.
For the engine air filter, you can mitigate this by purchasing a K&N air filter (or similar), which is washable, rechargeable, and reusable. They also have a million mile warranty. You need to buy recharge kits every 8-10 recharges, but it's by far cheaper in the long run. These types of filters are made of cotton and are backed by a wire mesh. They can also help to straighten out the airflow which can help improve performance (to a certain extent).
I have not heard of a reusable in-cabin air filter like the K&N I mentioned above, though they might exist.