Imagine that you're in a large room - say, 50 m x 50 m - which is so filled with people that on average you only travel about 1 meter before colliding with somebody else. Everyone on the left side of the room has a red hat and everyone on the right side of the room has a blue hat. The only way people move is by frantically sprinting into each other and bouncing off in random directions - that is, there is no coherent motion or intent behind their individual trajectories. How long will it take until the different colored hats are more or less uniformly distributed throughout the room?
The key question is not how fast any individual is moving at any one time, but rather how rapidly the group of red hats will spread out under the influence of chaotic, random jostling. The same is true of the atoms in your question. This kind of motion is called diffusion.
I should also add that your question body asks a somewhat different thing than the question title. When the gas comes to thermodynamic equilibrium, it isn't because the energetic molecules diffuse throughout the system, but rather that their energy does. Energy is transferred between particles in every collision, so even if a particular collection of molecules remains localized in one area for a while, their energy can spread throughout the gas.