1) The Pokemon Professor Program is a program in which the more knowledgeable members of the Pokemon community are recognized for their devotion and know-how of the game, and is a way for them to give back to the game (Pokemon) they love and wish to support. Generally, those who are in the program are well-versed with card rulings, and are more likely to be selected for judging or staffing at tournaments than non-Professors would be.
Note - the requirements to join the program is that you must pass the Pokemon Professor Exam, a 40-question exam that covers card rulings, tournament procedures, and judge know-how. To be eligible for the test to begin with, you need a Player ID (POP ID) and be at least 18 years of age; a passing score on the test is 32 out of 40 or more - or 80% and up.
2) Why? I wanted to find a way to give back to Pokemon - since I love the franchise, the creatures, and the trading card game as a whole, I wanted to find a way to contribute more and put my knowledge to good use. I'd judged one tournament before I was 18 (at the very start of January 2009) and have been judging regularly ever since. Granted, the perks are really nice (I'll cover those in question 3), but I was in it primarily for the aspect of helping out.
3) In addition to the Professors being respected members of the community, Professors also gain "Professor Points" based on how many leagues and tournaments they help to facilitate - for leagues that implies being a league leader or owner; for tournaments that implies being a judge or TO. These points can be traded in for various goodies at the Professor Store, which range to dice to special foil cards to the coveted Professor Lab Coat.
4) Helping to educate people. I honestly find that sharing my knowledge with others is a great thing to do. I'm not exactly in it for any personal glory - my goal is to help others, not to elevate myself for any reason. I've found the job to be richly rewarding, yes, and I've found judging more fun than playing (and it's not just because I get paid in packs), so all in all, these past 3 years have been some of the best years of my life - and I don't want to quit any time soon.