Let me try my explanation...
Using my PSH-1 conversion gun, which requires around 10-11 cfm, I use my "big" compressor (where big is a 13 gallon tank but runs on 120 volts, so cannot be over 1 1/2 honest hp) but that compressor only produces about 4-5 cfm. NO PROBLEM. Even though I use air twice as fast as the compressor can produce it, the tank gives me the air I need. As long as the trigger is pulled less than half of the time, the compressor can keep the gun happy and the air tank sort of full. The compressor may be running close to full time while I draw air half time. I only remember running out of air once, and it was after 5 minutes of almost continuous spraying (and may arm was as tired as the compressor).
Second trick. I run my air hose at 100-120 psi, so I can get a lot of air through the small hose. At the end of the hose is a pressure reducer - as the pressure is reduced to 20 psi going into the gun, I get 5 times as much air volume as I started with. Boyle's law says pressure times volume is constant (constant temperature and other constraints, but close enough for discussion). So when I go to 1/5 the pressure, I get 5 times the volume. A small air hose is okay if the pressure in the hose is high, and the pressure is reduced after the hose. If I had to run 20 psi from the compressor to the gun, the standard air hose would constrain the air flow - you would want a bigger hose.
Third trick. Most HVLP conversion guns take about 20 psi in, but HVLP legally is required to have no more than 5 psi at the tip. No problem. As the 20 psi air goes through the gun, the air path gets larger, the air expands, and the pressure drops (Mr. Boyle again). But the air pressure police are happy with 5 psi at the tip. If you decide to run 30 psi into your conversion gun, your tip pressure may rise above 5 psi, and you may be breaking the law (in about half the states), but you may like the results better - depending on the gun, maybe better atomization, at the cost of more overspray. The CAT Jaguar gun even has two spray caps... one that meets HVLP requirements, and another that works slightly better, but with a tip pressure just over the 5 psi limit.
So do you need a 1 1/2 hp compressor with a 13 gallon tank to run an HVLP gun? NO. I know a professional furniture restoration "artist" who uses his hot dog compressor - typically about 4-6 gallon air tank. The only problem is that he has to pause occasionally when spraying to let the compressor catch up. (He is almost as old as I am, so his arm may need to rest too). I have a tiny compressor that I use to install mouldings on bookcases - just shoot a few brads at a client site. It probably has a 1 gallon or smaller tank. (I have to fill the air tank twice to pump up a car tire). I occasionally spray with it for touch-up work. It works just fine, but with the tiny tank and low volume compressor, I can't spray much before I need to wait for the compressor to catch up.
Now the second subject... air filter
If you have a oil lubricated air compressor (quieter and longer lasting than an oil-less compressor), a tiny amount of oil will escape into the sompressed air, which is not good for any finish, and especially bad for water based finishes. You can put a filter by the compressor, but if any oil gets into the hoses, they will never be perfectly clean. The recommended solution is a filter near the spray station. Jeff Weiss recommends the disposable Grainger 6ZC63. One of the users of that filter recognized that the primary contaminant was water, so the filters could be dried out and reused.
Third subject ... air tools.
I had some automotive type air tools (air wrenches, etc.) and they worked fine with my cheap compressor, but they made it work hard. I don't do a lot of mechanical work (I like wood) so I sold the air tools.
At one of the big woodworking trade shows, a salesman decided I needed an air sander. After cutting through the hype, the bottom line was that I needed a 5 hp compressor to run one sander. If I were in an explosive environment (like an auto body shop with flammable solvents and paints) it would make good sense. In a wood shop, I looked at 4-5000 watts electricity for the compressor to run the air sander, compared to about 300 watts to run my big 6 inch Festool electric sander. For 1/15 as much power I can handle the slightly higher weight of an electric sander, thank you. But people of good character love their air sanders.
A big compressor is recommended because it can put out the high volume (CFM). In most cases, large compressors don't put out much more pressure (PSI) than a little pancake compressor, but they typically put out a much higher volume of air. But I'll bet you already knew that. 
