A multinomial distribution is one where every observation in the dataset is taken from one of a limited number of options. For example, in the race census data, race is a multinomial parameter: it can be one of seven options. If the census were a sample, how good of an estimate of the population would the ratios of the race observations be?
Bayesian methods work by updating a prior probability distribution on the data with more data. For multivariate data, the Dirichlet distribution is commonly used. The Bayesian process observes how many times each option is seen and returns an estimate of the ratios of the different options from the multimodal distribution.
So in the case of the census race data, this algorithm looks at the ratios from a sample and updates the prior distribution from those values. The output is a belief about the probabilities of those ratios in the population.