What does the quote,
"If a man can write a better book or preach a better sermon or make a better mousetrap than his neighbor, even if he builds his house in the woods, the world will make a beaten path to his door"
The meaning is pretty much what he writes. A short paraphrase is
If a person can do something better than others, then other people will seek them out.
If a man can write a better book or preach a better sermon or make a better mousetrap...
Those are just examples. He means in general if someone can do something well.
than his neighbour
than other people around him, not just those in his neighbourhood.
even if he builds his house in the woods
even though he lives somewhere remote and hard to get to
the world will make a beaten path to his door
people will put in a lot of effort to get to him. Specifically this refers to making a path through the wood, often done by beating down the shrubs and plants until they can be walked over.
It's worth noting that the phrase 'build a better mousetrap' has passed into the English language as a phrase meaning 'make something better than everyone else', thanks to this quote.
This wasn't actually ever said by Emerson - according to Wikepedia the actual quote is:
"If a man has good corn or wood, or boards, or pigs, to sell, or can make better chairs or knives, crucibles or church organs, than anybody else, you will find a broad hard-beaten road to his house, though it be in the woods."