Above is a screenshot (from this page) that is showing how to compute the probability of the observed data given a Dirichlet prior. I don't see how they got from the expression in blue to the one in red. Isn't that a multinomial expression that requires a term that specifies how many ways you can observe that data? Something like this:
The unary representation, or creation of "dummy variables", seems to reflect the passage that you showed in your example.
https://en.wikipedia.org/wiki/Dummy_variable_(statistics)
I'm assuming that $y_i$ ranges from 1 to k. The probability that $y_i=j$ equals $p_j$.
The passage that you emphasized there corresponds to the categorical distribution
https://en.wikipedia.org/wiki/Categorical_distribution
and the Dirichlet is the conjugate to the categorial distribution
https://en.wikipedia.org/wiki/Dirichlet_distribution
In your question, the Dirichlet distribution appears in the term in the left just before the one that you marked in red.