Start in this sense means an initial advantage. I have an advantage over you at doing something quicker if I start earlier or start further along in the task, and this can apply figuratively to anything that gives someone an initial advantage. (Compare "head start").
The saloons would have been a competitor for the attentions of the clergyman's potential flock. They would likely have been a competitor in other ways depending on how liberal the clergyman was or was not when it came to the topic of alcohol, and on how many things happened in them that even a clergyman who tolerated liquor would object to.
By way of means (among other senses), "for the purpose of". Which is to say this rent-free arrangement was offered with the intention of providing this advantage.