The condition relates activity coefficients in liquid phase to total pressure and the vapour pressures of pure components.
Azeotropes can form only when a mixture deviates from Raoults law, the equality of compositions in liquid phase and vapor phases,  in vapour-liquid equilibrium  and Daltons law the equality of pressures for total pressure being equal to the sum of the partial pressures in real mixtures.
In other words: Raoults law predicts the vapor pressures of ideal mixtures as a function of composition ratio.  More simply: per Raoults law molecules of the constituents stick to each other to the same degree as they do to themselves. For example, if the constituents are X and Y, then X sticks to Y with roughly equal energy as X does with X and Y does with Y. A positive deviation from Raoults law results when the constituents have a disaffinity for each other – that is X sticks to X and Y to Y better than X sticks to Y. Because this results in the mixture having less total affinity of the molecules than the pure constituents, they more readily escape from the stuck-together phase, which is to say the liquid phase, and into the vapor phase. When X sticks to Y more aggressively than X does to X and Y does to Y, the result is a negative deviation from Raoults law. In this case because the molecules in the mixture are sticking together more than in the pure constituents, they are more reluctant to escape the stuck-together liquid phase.
When the deviation is great enough to cause a maximum or minimum in the vapor pressure versus composition function, it is a mathematical consequence that at that point, the vapor will have the same composition as the liquid, resulting in an azeotrope.
The adjacent diagram illustrates total vapor pressure of three hypothetical mixtures of constituents, X, and Y. The temperature throughout the plot is assumed to be constant. The center trace is a straight line, which is what Raoults law predicts for an ideal mixture. In general solely mixtures of chemically similar solvents, such as n-hexane with n-heptane, form nearly ideal mixtures that come close to obeying Raoults law. The top trace illustrates a nonideal mixture that has a positive deviation from Raoults law, where the total combined vapor pressure of constituents, X and Y, is greater than what is predicted by Raoults law. The top trace deviates sufficiently that there is a point on the curve where its tangent is horizontal. Whenever a mixture has a positive deviation and has a point at which the tangent is horizontal, the composition at that point is a positive azeotrope. At that point the total vapor pressure is at a maximum. Likewise the bottom trace illustrates a nonideal mixture that has a negative deviation from Raoult's law, and at the composition where tangent to the trace is horizontal there is a negative azeotrope. This is also the point where total vapor pressure is minimum.