Split and balance half reactions.
A half reaction accounts for one side of an electron transfer. The electron count is fixed by how far the oxidation state moves, multiplied by how many atoms move — permanganate going from Mn(VII) to Mn(II) is a five-electron reduction, while dichromate reduces two chromium atoms from VI to III and therefore takes six electrons. Once the electron count is known, the ion-electron method balances oxygen with water and hydrogen with protons, adding hydroxide at the end for basic solution.
Electron count from oxidation state change
electrons = |state after - state before| x atoms changing; positive change is oxidation, negative is reduction
Free elements are 0, monatomic ions equal their charge, oxygen is normally -2 and hydrogen +1, and the states in a species must sum to its overall charge. In MnO4-, four oxygens at -2 total -8, so manganese must be +7.
Because H+ cannot exist at appreciable concentration in base. The standard trick is to balance in acid first, then add the same number of OH- to both sides, which converts each H+ into water.
It is what makes the two half reactions combinable. Multiply each half reaction so both carry the same electron count, then add them so the electrons cancel — that is how the full redox equation is built.