Can someone please make me understand how balancing equations works?

Balancing chemical equations or mathematical equations?
Your question is not specific enough. I will assume you mean chemical because mathematical equations are straight foward enough.

Here is a simple equation:

2H2 + O2 -> 2H2O
This is the formation of water from hydrogen and oxygen. You can see there are two moles of Hydrogen and one mole of oxygen gas on the left hand side. On the right there are two moles of water molecules. To see that this balances just count up how many of each ATOM is on each side
Side A: 4H + 2O
Side B: 4H + 2O
You can see this adds up.
So all you have to do is look at the subscript(if any) for hydrogen there is a small 2 to the bottom right, this means two hydrogen atoms per molecule, this needs to be multiplied by the number all the way to the left (2) to get 4 hydrogen and so on.