Here's one simple explanation:
When you breathe in you inhale 21% oxygen and 79% inert gas (that's from your basic OW scuba classes). As you know, what your body needs to live on (the fuel) is the oxygen. But of that 21% you inhale your body only needs about 3% to sustain life. So, when you exhale, you are actually thowing away about 18% of the oxygen you could have used (metabolized) as fuel. Let me rephrase that: It means when you are under water on open-circuit scuba, every time you exhale you are thowing away about 18% of the oxygen you could have used to dive with.
A rebreather lets you keep that 18%, and keep breathing it (why it's called a rebreather) until all of the oxygen is used up. Basic division says if what you inhale contains 21% oxygen and you only use about 3% of it, then you should be able to get about 7 breaths on a rebreather (21 divided by 3 = 7) for every breath on open-circuit scuba. See an advantage - staying down 7 times as long as you can with the same amount of air.
Naturally you have to add oxygen as you begin to burn it down.
One last thing: The byproduct of buring/metabolizing oxygen is carbon dioxide, which is not something you want to be breathing. A rebreather uses a filter to absorb that carbon dioxide. Most rebreathers are limited as to how long they can stay down by how much carbon dioxide their scrubbers can absorb.
From there it's all a matter of which rebreather you select. They all work basically the same way, but are configured differently and may operate in different ways, as has already been discussed in earlier posts.
Hope that is helpful.