Syncopation is an often-used and less often-understood word. I take syncopation to mean nothing more or less than this: Playing a melody note at an unexpected time. Or some people would call a syncopated rhythm one that hits not just on the beats and offbeats, but anticipates or falls between main beats.
For my money, the best music is made by people who feel and hear the music in their heads and bodies almost simultaneously with thinking about it and actually doing the playing. What it boils down to is you have to *think* syncopated to play syncopated. I suppose there are ways to note that on a page for someone reading music. But I prefer hearing music that is felt as it's played, not just something that's recited from a page.
How to feel syncopation? I would say to check out musicians you like who are noted for that. Earl Scruggs has a way of playing tricks on listeners by syncopating melodies, almost as if he's playing a game. Reggae music has a lot of syncopation, and quite a few forms of black music use looser, more complex rhythms than most white forms. Syncopation was also a trademark of Frank Sinatra.
Have just..... a heck of....a TIME. (syncopated writing??).
Pete