The title really says it all: Why is this case? A "Feynman type" answer would be really appreciated as I'm more of a layman that a physicist.
Answer
The answer to this question is quite intuitive when you think about what pressure is: a force per unit area. In a high pressure zone, particles experience a high force, and in a low pressure zone, they experience a lower force. The high force "overpowers" the lower force, pushing the particles from the high pressure zone to the lower pressure zone.
You can also think about this from a statistical thermodynamics standpoint. Consider the following thought experiment: You have two containers, one with high pressure gas and another with lower pressure gas. The high pressure container contains a lot of particles per unit volume (that is, it's relatively "full"), and the lower pressure gas contains few particles per unit volume (it's relatively "empty"). When the two containers are put side to side and gas is allowed to flow, the "full," high pressure container will lose particles to the "empty," low pressure one, causing particles to move from high to low pressure again. Note that this effect is purely due to random movement of the particles. The equilibrium position, of equal particle densities everywhere, is simply the one that has the largest chance of happening (and an overwhelmingly large chance at that) in the long run.
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