It's a combination of his cadence and timing. If you really want to analyze it, try to play his voice on a piano or something with a metronome on, write it out, and study it. This helps you see a lot, like the melody he's making with his voice, the timing of the notes, and also where he places silence which is a huge part in making something bounce.

(I was the) drama club kid, I run

Notice that he actually starts the verse at the end of the previous bar with "I was the". This by itself creates extra swing in his flow, because he's starting his rapping before the bar actually starts.

Next, look at his timing on his words. They're not all equal obviously, but it's nice to see what he's doing at the fundamental level. "drama" is two sixteenth notes, "club" and "kid" are both eighth notes. He prolongs the timing of some words like "run" and "fun" and also takes some silence before some bars to catch it off beat.

Also pay attention to his cadence. Rapping is basically singing, he's hitting the same notes when he says "drama club kid", "where the fun did", "nuts itched", and "fuck shit". If you play it out on piano you can notice this right away. You can say he just made a melody, and then added words on top of it. Making a catchy melody is hard, that's why a lot of people suggest to hum or acapella through your beats to catch different flows/melodies.

Experiment with slicing the delivery and timing of your lyrics.

Overanalyzing it a bit, but it's nice to know what's going on fundamentally. It's basically just about switching up the flow by using silence, different timings, repeating melodies/motifs, etc. Hope that helped a bit.