This is count data. That rules out uniform (continuous or discrete) as well as normal. The only possibilities based on data type are the Poisson and the Binomial. The binomial does not seem appropriate because this is not the number of outcomes for a fixed number of independent experiments where each of n people can have their bone broken with the same probability. The Poisson fits because it represents certain rare event hypotheses and is a number of broken bpne events observed over a given interval of time (ome football season). It is not clear that the Poisson is the best model but it is better than the other choices. The number of college football players in finite so there is a fixed finite limit to the number of events when technically the Poisson has no limit.
If someone argued for the binomial because there is a fixed finite number of players available at the beginning of the season that are at risk for injury from a borken bone on any individula play and the plays are independent. Then the trial could be considered the plays and with 22 players available on the play may indicate a constant probability that a broken bone will occur on the play. The problem with that argument is that the number of plays in a season is not known and should be considered random (unless you look ay this conditional on the number of plays that actually occurred during the season). The other issue is that plays differ in the injury risk. Real life situations can always lead to arguments against a model but the shear randomness and rarity of bone breaks points to the Poisson as the best model of the 4 choices.