It means that if you become rich and famous and start to think you are better than everyone else, at least you will have me to remind you that you are not better than everyone else. This knowledge will keep you humble and prevent you from developing an over inflated ego and treating people poorly, especially those less fortunate than you. Treating people poorly could cause your rich and fame to go away, to crumble. Thankfully, again at least you have me to remind you of your humanity, your warmheartedness, and your friendship. This is why you can count on me to keep you humble.

"lest your head should swell" means "should your head get big/should you have a big ego". If your head literally got big, it would crack and fall apart (crumble). Being humble is the opposite of having a big ego. So basically that verse means "if you start to get a big ego, I will be here to make sure you stay humble".