The calculation sums, for every strike, what all outstanding calls and puts would be worth if the underlying settled there, and reports the minimum. The theory attached to it is that market makers hedge in a way that nudges price toward this level into opex.
The calculation is real; the theory is mostly folklore. It assumes dealers are net short all options and have both the ability and the incentive to steer a market, neither of which is generally true. The underlying phenomenon — pinning driven by dealer-gamma near heavily traded strikes — is real, but max pain is a poor proxy for it.
Example: XYZ's expiring chain gives a max pain strike of $50 with the stock at $52. The prediction is a drift toward $50. The honest reading is that $50 has the largest open-interest and therefore the strongest hedging flows near it, nothing more.
Related: pinning, opex, dealer-gamma, gamma-exposure