Adding 'out' to a verb generally makes the action seem more complete. For example, if you say 'John helped me when I was in trouble', we understand that John gave you some help. But if you say 'John helped me out when I was in trouble', we understand that he probably solved your problems completely.

"Help me out" is usually asking for help that is not urgent, perhaps money, food, a ride to somewhere, or support in a social situation. "Help me" is usually used for more urgent help. If I was drowning or stuck under a rock, I would say "help me", not " help me out". If I wanted a loan I would say "help me out" not "help me". Unless I was about to starve without the loan, then I might say "help me". If some one was chasing me with a gun I would say "help me", not "help me out". If I was asking my child to carry the groceries with me, I could say "help me carry the groceries", or I could say "help me out with the groceries". The first one is more of an order, and the second is more casual.