braggarts (noun)  a loud arrogant boaster

“Never listen to what people tell you about old New York, because if you didn’t witness it, it is not a part of your New York and might as well be Jersey. Except for that bit about the Dutch buying Manhattan for 24 bucks — there are and always will be braggarts who “got in at the right time” City Limits by Colson Whitehead paragraph 5

This part of the passage means that if you didn’t witness it yourself then it is not true and it cannot be a part of your New York. However, there will always be loud and arrogant boasters that are always bragging about how they made such a profit or a good event that happened to them.