Evolutionarily, why do male rats and horses lack nipples?
Male mice lack nipples too. Mice are frequently used for embryonic research as they are small and reproduce quickly. It is thought that male mice do develop nipples, but that they regress during development (Wysolmerski, 1998). 

In general, it is thought that mammalian organisms develop as females by default when there is no male (Y) chromosome present (Hughes, 2004). Notably, X_ individuals and XY individuals with certain Y-chromosome deletions also develop as females. Basically, it is testosterone that suppresses female features. Hence, also boys with testosterone production, but without the appropriate receptors also develop as females.

Hence, men have nipples because females do (Simons, 2003).

The development of nipples is a pretty complex process (Wysolmerski, 1998). Disruptions in the process cause regression. Indeed, in mice, the tissues typical for nipples do develop in males during embryogenesis, but degenerate within a few days, leaving no trace of nipples at birth.

Hence, given the mouse example, I think that every mammalian male species develops by default nipples somewhere in development like mice, but that development is halted at various stages across species. The later in development it halts, the more it resembles female nipples. When it is stopped very early after gestation, they regress completely, like in mice.   

Why is there a difference between species as to when the nipple development halts? Because the nipple developmental sequence is complex (Wysolmerski, 1998), and because evolution typically prefers losing things over gaining them, it is likely that by happenstance one of many crucial genes was lost in males, but selective pressure was simply too small to be of any significant advantage over males with slightly better-developed nipples. But from here on, it is guessing.

References
- Hughes, N Engl J Med (2004); 351(8): 248-50
- Lawrence, Nature News, August 1999
- Simons, Sci Am, September 2003
- Wysolmerski et al., Development (1998); 125: 1285-94