As Yashas said, the standard "math-free" explanation is that the space in between the particles stretches, so that the distance between the particles is growing faster than the speed of light. Now we have to explain why this doesn't contradict your statement, "the particles never exceed 186,000 miles per second".
The problem is we need to specify what this speed is relative to. In special relativity, you can compute the speed of any object relative to any other object (e.g., the speeds of the two particles relative to each other). However, this is only possible since spacetime in special relativity is uniform. Every point is the same, so anything at any two points can be compared.
In general relativity, spacetime is no longer homogeneous, and as a result it is impossible to compare vectors, such as velocities, at two different points. You can say that a particle has just moved past you at some speed, but it's meaningless to calculate the relative velocity of two distant particles.
As an analogy, suppose we stand at two different spots on the equator, both pointing North. If we wanted to compare the directions we're pointing, we have to meet in the same spot. If we meet on the equator, our directions will agree. If we both march up to the North pole and compare, they'll disagree! This happens because the Earth's surface is curved. For identical reasons, it is impossible to compare distant velocities in curved spacetime. There's no unique way to do it.
As such, the fact that the distance between them grows faster than $c$ doesn't contradict relativity. Nothing is ever traveling faster than $c$ according to a local observer.