I think I see a wall separating the savages from an area from what looks like califrnia to texas..
Most of the western coastlines are conjecture. The map shows part of the South American east coast, the Central America east coast, Gulf coast, Florida, Cuba and other parts of the Caribbean, and part of our East Coast. They could map the coast lines that they could see and could plot their latitude. Longitude was a guess, and coasts that they had not visited were fictional.
How did they know that water surrounded the new continents? Deduction, people had been to China and looked upon the Pacific Ocean to the East I assume.