the one where you can switch them is understandable, the one where you can multiply through by a constant is easy, but the one where you add the multiple of one row to another row to get a new row is weird. I've found that it's similar to solving linear equations by adding them together ex
3x + 4y = 2
-3x - 2y = 1
2y = 3
but back when I was learning that I wasn't really concerned with proof. I guess I don't even see why the above works.