I understand that for certain datasets such as voting it performs better. Why is Poisson regression used over ordinary linear regression or logistic regression? What is the mathematical motivation for it? 
Poisson distributed data is intrinsically integer-valued, which makes sense for count data. Ordinary Least Squares (OLS, which you call "linear regression") assumes that true values are normally distributed around the expected value and can take any real value, positive or negative, integer or fractional, whatever. Finally, logistic regression only works for data that is 0-1-valued (TRUE-FALSE-valued), like "has a disease" versus "doesn't have the disease". Thus, the Poisson distribution makes the most sense for count data.

That said, a normal distribution is often a rather good approximation to a Poisson one for data with a mean above 30 or so. And in a regression framework, where you have predictors influencing the count, an OLS with its normal distribution may be easier to fit and would actually be more general, since the Poisson distribution and regression assume that the mean and the variance are equal, while OLS can deal with unequal means and variances - for a count data model with different means and variances, one could use a negative binomial distribution, for instance.
