Wikipedia articles on statistics are great. I didn’t know the Dirichlet had a “balls in an urn” explanation. But I’m not too surprised as everything can be explained by balls in urns.
Consider an urn containing balls of K different colors. Initially, the urn contains α1 balls of color 1, α2 balls of color 2, and so on. Now perform N draws from the urn, where after each draw, the ball is placed back into the urn with another ball of the same color. In the limit as N approaches infinity, the proportions of different colored balls in the urn will be distributed as Dir(α1,...,αk ).
Note that each draw from the urn modifies the probability of drawing a ball of any one color from the urn in the future. This modification diminishes with the number of draws, since the relative effect of adding a new ball to the urn diminishes as the urn accumulates increasing numbers of balls. This “diminishing returns” effect can also help explain how large α values yield Dirichlet distributions with most of the probability mass concentrated around a single point on the simplex.