The Leave of Absence (LOA) is simply a break in a student's enrollment. The FSA Handbook and CFR are vague on some aspects of the LOA, but are quite specific on other aspects. But what happens when you come back from the LOA?
Well, you should know certain things. You can't get charged while on the LOA. If you took the LOA at the beginning of your semester/quarter, then when you return you will just get charged for your next enrollment period. If you were in the middle of your semester/quarter, then when you return, interesting things can happen. You will not get charged for your next enrollment period until you complete it (since you started one but didn't complete it). Any FA that has already come in before the LOA began will not come in again until you complete the enrollment period, and any FA that hasn't come in yet can come in then. This is why if you have money coming to you, and you received it before the LOA, then there's a good chance you won't get more until you complete the enrollment period that you return for. Each situation is a little different, so you need to check with the FA office to see how it affects your situation.
The other issue is your academics. For term-based schools, you must be allowed to complete the coursework that you began before the LOA. Most people interpret this as you should be in the same classes as when you took the LOA. For nonterm-based programs and clock hour programs, you don't have to complete the same coursework you began before taking the LOA. In those programs, you aren't necessarily (depending on the school) in regular enrollment periods, so you are eligible for funds near the beginning and at the halfway point. When you hit the halfway point, then you are eligible for the next disbursement. Because of this, you take your hours or do your coursework remaining until you hit the halfway point.
The Leave of Absence is a strange and sometimes complicated situation and some schools won't do them for that reason. They are hard to explain, and they affect not only your FA, but your academics, your graduation date, and the possibility of your having to be dropped in some cases. In the end, if your school does offer LOAs, you should only consider it as a last resort, and even then make sure you understand the consequences of taking an LOA.