In one line
A payoff diagram is what a position is worth at expiry for every level the index could settle at, and the breakevens are the two levels where that value crosses zero.
How it works
At expiry the arithmetic is exact and there is no model in it. Each contract is worth its intrinsic value and nothing else, so the payoff is a set of straight lines that change slope at the strikes. The premium paid or collected shifts the whole picture up or down. Breakevens are where the line crosses zero, and because the payoff is piecewise linear they can be found exactly rather than approximated — provided the search is seeded with the strikes, or it can step over one.
BEFORE expiry is a different question and this desk answers it with a different function that is named differently. There the position is worth whatever the market charges for its legs, which depends on time left and on implied volatility, so it is a model rather than arithmetic. Where the model cannot be evaluated the desk returns nothing rather than quietly handing back the expiry curve, because a curve that silently became a different claim under the same label is worse than no curve.
THE MOST IT CAN LOSE IS READ FROM THE LEGS, NOT FROM THE EDGE OF THE PICTURE. A written call has no upper bound, and printing the loss at the right-hand edge of whatever range was drawn would be the most dangerous number on this site. Unbounded is reported as unbounded.
What it is not
A payoff diagram is not a probability distribution. It says what each outcome is worth, and says nothing at all about the chance of any of them. Two positions with identical pictures can be very differently exposed, because the picture does not carry how much time is left or how much movement is priced.