I missed “Treme” the first time around, but have lately discovered and began binge-watching it. It’s eminently binge-watchable, IMO. But I am puzzled over this bit of musician-speak that has come up every so often on the show.
So, musician dopers, what exactly does it mean to clap on the one and the three? I can guess that it has something to do with the beat and keeping time. Based on the context the line is delivered in, it is just about the worst burn one musician can give to another.
An explanation:
http://boards.straightdope.com/sdmb/showpost.php?p=4983000&postcount=7
For an example:
http://boards.straightdope.com/sdmb/showpost.php?p=18675790&postcount=8
Look around you at the next concert where people are attempting to clap in time with the music. See those few people who are not clapping along with the others? Those are people with no sense of rhythm who are clapping on the downbeat. Many musicians discourage audiences from clapping with the music, or one member of the band will do an exaggerated clapping motion to get everybody on the 2 & 4, since for some reason people feel the need to do this. There are also musicians (I think Harry Connick is one)who will throw a hitch into the piece to make it sync up with a particularly musically-impaired audience.
Do you know the fight song, “Ramblin’ Wreck from Georgia Tech”?
There will be folks who start clapping at “Ramblin’”. There will be folks who start clapping at “Wreck”. That first group is much more likely to be white than black.
This isn’t to say that most white people are going to be in that first group. I think listeners of modern American pop music are inclined to clap on the 2 and 4, because that’s where the emphasis tends to be placed in modern American pop music. But occasionally you’ll see people deviate from this, and those folks tend to be white. Thus, the joke is that only very white people clap on the 1 and the 3.
There’s a stereotype of white people not having any rhythm. This isn’t true, IMHO. It’s just that white people tend to have a different rhythm than the rest of us.
My experience is that usually most of the audience claps on the downbeat, to the annoyance of those few who have a sense of rhythm.
Doesn’t this paragraph contradict your first paragraph above? 
There’s this video where Justin Bieber stops the song and chastises the audience for clapping on the wrong beat. 
Funny thing: for a lot of songs, if I try to clap on the downbeat I find it really difficult.
And ask my wife: I ain’t got no rhythm.
But clapping on the “right” beat seems so natural to me for pop music it’s hard for me to understand how people get it wrong.
And I was even raised in white-people-clapping church.
Brain bleach?
Huh…I thought they were adorable.
Nothing’s more square than clapping along to music, whichever beat of the bar you’re hitting.
While that’s probably true in general, where would Flamenco Dance, Seville, Spain be without the clapping and such? 
Watch this repeat of the Harry Connick videowith commentary. It explains this phenomenon quite clearly.
I saw the Flying Burrito Brothers live in the 70s, for a concert that was being recorded for a record. After they played a song, they said, “It’s OK to clap along, but clap on the beat” and then played the song again.
I often clap on 1 and 3, not because I don’t know it, but I like the sound of a clap that isn’t on the beat.
I have no problem with people clapping on beats 1 & 3 . . . provided they’re not in 4/4.
“If you’re white, clap out of time”.
Saw a black comedian with a guitar who worked up a bit on this.
But yeah, we stupid white folks just cannot clap on time.
The link in your post was a great explanation/example. At the start of this thread I didn’t even know what it was talking about. With the Harry Connick explanation (with foot notes) I wasn’t sure there was a difference. But this simple explanation and a capella demo made it clear.
I love listening to music, but can’t play a note. When my kids were in grade-school band and would ask me questions about their music homework I felt like an illiterate parent. Now to see if I can tell the difference in real-life. Learn something new every day.