Pick a half-reaction and walk the recipe one move at a time. The tally board counts atoms and charge after every step, so you can see exactly when each side comes out even. Then flip to base and watch hydroxide do its job.
Water is the only thing you can add that brings oxygen without bringing charge. Once oxygen is even, hydrogen is the only atom left, and H+ carries it. Electrons are last because they are the only piece that changes charge without changing any atom count.
In base you add the same number of hydroxide ions to both sides. That keeps the equation true. On the acid side each pair becomes water, and the extra water cancels out.