He is just playing with repeating sequences that don't fit nicely into certain time signatures.
I'll try to explain some basics here. The most common time signature is 4/4. That means four beats:
1 - 2 - 3 - 4
The most basic drum pattern would be
B - S - B - S (Bass and Snare - think AC/DC)
Each beat can be divided by 4 (sixteenth notes) and is counted out as:
1 e & a 2 e & a 3 e & a 4 e & a
You can see that is 16 subdivisions.
So Mangini is talking about patterns of 17 and 19, which don't divide nicely into 4/4 (16 subdivisions)
In other words, when the 4/4 ends (and begins to repeat back on 1), the 17 pattern still has 1 more subdivision. The 19 pattern has 3 more subdivisions.
So the second time around, the 17 pattern would begin on the e of 1
1 e & a 2 e & a 3 e & a 4 e & a
third time around, the & of 1
1 e & a 2 e & a 3 e & a 4 e & a
fourth time around, the a of 1
1 e & a 2 e & a 3 e & a 4 e & a
fifth time around, it is back on the beat, but the 2, not the 1
1 e & a 2 e & a 3 e & a 4 e & a
Etc. etc.
It is definitely not an easy thing to do as it requires the ability to think about two different beats at once (hence, polyrhythms)
And the 3 layer means putting the 4/4, 17 pattern and 19 pattern together all at once.
This is why he says it isn't meant to be musical. The idea is he practices this type of stuff so it comes out more naturally later on in life.