"His dad explained that the man was very successful in business because he had no competition—he rendered dead horses." More context is needed, but I think it is possible "he rendered dead horses" is used as a metaphor here (and not meant to be read in its literal sense). "Rendering dead horses" is a very unpopular job. It involves converting horse flesh to useable products but it is a very toxic and smelly process. Very few people would want to do it - you would have little competition. I think the text is simply EMPHASIZING that the man was successful in business because no one else wanted to do whatever he did. He was successful in business because no one else wanted to do it (like no one would want to render horses).

Most of Zoë's answer was correct, but I disagree that 'to render' means 'to offer a service'. It is quite specific, and means something like 'to make products out of dead animals'. See https://en.wikipedia.org/wiki/Rendering_(animal_products). It's also not limited to the US. People in the UK would refer to such a person as a 'Knacker', which is similarly difficult to understand without context! ;)

This is a little hard to understand out of the context of the whole text this paragraph comes from. 'To render', in this case, means to offer a service. Rendering facilities throughout the U.S. offer varied services including pick-up services for animal carcasses and onsite euthanasia. So it's guy who picks up deceased animals and transforms them in to different things. The process produces a variety of products including tallow, lard, animal feed, protein meal, cosmetics, and mechanical lubricants. Interesting! Thanks for sharing.