This is a common source of confusion because many folks figure that a uniform distribution would be very ordered.
The thing's that things aren't becoming uniformly distributed; that's just a macroscopic view! What's actually happening is that the macroscopic view is becoming increasingly less informative, i.e. uncorrelated with the information of the system. By the time complete macroscopic uniformity is reached, the observer has completely lost their ability to guess what actually exists at any given point in that space.
For example, consider a glass of water with a block of salt to be dropped into it. Once the salt is dropped into the water, it'll dissolve and evenly spread around. That seems more ordered and uniform, huh? But, it's not!
Before mixing, you could say if there was a water particle or salt particle at some randomly selected point in the glass.
As mixing occurs, the amount of information you have about relative probabilities of whether there's a salt particle or water particle at any given point become less certain, though you can still make a better-than-no-information guess at the odds.
After complete mixing, you have no idea beyond saying that there's a single probability throughout the glass equal to the portion of the particles of the selected type.
This is, the structure (order) that existed before mixing has now been broken down into random chaos. And that's pure entropy.