Android - How RAM usage effects battery drain?

I doubt that there is a direct correlation between battery consumption and RAM usage. The hardware doesn't know which RAM cells contain 'used' data and which not. So there can be no difference in battery consumption on that level.

But I think that one could say that unnecessarily killing Apps causes a few extra CPU cycles when those Apps have to be re-started and therefore "uses more battery".

And yes, you definitely should let go of your app-killing habit. In most cases, Android is capable of managing the system's resources quite well and you shouldn't bother with manually killing Apps. They definitely won't drain more battery if they just idle around and consume some RAM. Of course, if those Apps are constantly using the CPU, they will drain battery (but this is usually a sign of a bug within the App or a badly implemented App).

Think of RAM like a paper notebook. You can write data into the book (with a pencil), and you can erase those data and replace them with new data, but the book's always the same weight. The book doesn't get any heavier, whatever you write in it. The same way, with current RAM technology, the battery use of the RAM is fixed, regardless of what (if anything) is stored in it.

From Android's point of view, it doesn't matter whether you leave an app with the home button or the back button. Pressing the back button doesn't kill an app or remove it from RAM. Swiping an app out of the 'recent apps' list doesn't kill the app or remove it from RAM. The only difference it makes is changing what the app shows you next time you launch it: the screen you were on before, or the main/first screen of the app.

Imagine you have a notebook (as before), and a big bookcase with lots of reference books. You can't carry the reference books around with you, so if you need to know about a subject, you have to copy the information out of a reference book into your little notebook. When you run out of space in the notebook, you use your eraser to wipe some information you don't need again, so you can copy more information into it.

All this copying is tiresome. So once you've copied some information, you want to keep it in your notebook for as long as you think you might need it. You keep track of what information in the notebook you're using right now, and what information you use often, and what information you haven't used for a while. When you want to make space, first you erase the information you haven't used for a while. If you haven't got any of that left, then you erase some information you use often but not right now. Only if you're really desperate for space (maybe there's one big subject that takes the whole notebook) do you erase the information you're using today.

This is exactly what Android's doing with your RAM. The big bookcase with reference books is like your phone's internal storage: you can't use it directly (because the books are printed, not written with pencil), so it's expensive (it takes time and energy) to copy data into your notebook (RAM) to work on.

A task manager app is like a schoolteacher who thinks your notebook looks untidy with all those pages of notes you're not using any more. The schoolteacher comes along every so often and erases whole pages of notes. This is unnecessary, because it doesn't make the book any easier to carry around: the book weighs the same regardless. Not only that, the schoolteacher doesn't know which information you use most often - he might have a vague idea, but he doesn't know as well as you do - so sometimes when he erases information, the cost is an extra trip to the bookshelf, which you wouldn't have needed otherwise.

Thinking about it this way, it's easy to answer your question. Using more RAM doesn't use any more power directly. A task manager app can make you use more power, because you're having to copy things from internal storage to RAM that you wouldn't have had to otherwise. In the same way, a single RAM-hungry app costs you twice: first you have to write all of that app's information in the notebook, and then when you finish with it, there's a lot of information that you had to erase (to make space), which you have to copy from the bookshelf again.