Those rhythms sometimes get called “2-step”. You can kinda hear the pulse… 1… 2… 1… 2… the deep heart of the rhythm is a tight kick on 1 and 9 (in a 16-step, 4/4 configuration), and very snappy snares on the backbeat (5 and 13). The kick on the 9 is optional, or can move to the 11.
You can work with 4/4 time, using swing. Or 12/8 time using “triplets”. You’ll want pattern lengths in multiples of 4 for 4/4 time, or 12 for 12/8 time. I’m going to write up the next bit assuming 4/4 time with swing, because it’s easier to work with on an Elektron that has 16 step/trig buttons.
The mid points between the on-beats (e.g. the 3, 7, 11, 13 in 4/4 time) are usually also “fixed” (not swung, not in triplet time), but you can play with that. Lay some hats here, particularly closed ones, or rims, cowbell & sticks. Or miss some. Or put quieter kicks or snares for quirky variations.
The weakest steps (2, 4, 6, 8, 10, 12, 14, 16) are the ones where you can make great use of swing. Use more hats, especially if you didn’t put a hat on the preceding “stronger” step. Also use quieter (“ghost”) kicks and snare hits in these positions.
Set swing around 55-60% (in my opinion going any higher makes the upbeats too pushy. Feel free to take other advice 
).
Get something groovy going with these guidelines, then break the rules for fills, variations and other musical punctuation.
Someone better at math than me can explain how to use 12/8 time to do similar feeling patterns.
In terms of sound design…
- kicks: usually short; wide descending pitch envelopes, some upper mid-range in the transients; “woody” kicks are quite nice here; definitely avoid the long boomy kicks, unless you’re making a statement
- snares: usually snappy, sharp, 808s, 606s made really short; use short claps instead; I don’t think “spreading” the snare/clap around the beat using micro timing would sound too out of place on some tracks, if you like that… but most garage rhythms go for “snap”
- some old tracks drive the snares and hats into distortion; but keep aim to keep it rhythmically tight, don’t let the fuzz stretch the sound out.
I’m not sure about compression tactics. I guess something that emphasises the “springiness”.
You could play around with cutting up breakbeats - using fragments longer than 1/16th, so you get more than one hit in the sample. This’ll let you make use of interplay between the straight and swung parts of the rhythms.
If you want to sound like Burial, use more breakbeats, use more mid-range, roll off top-end, think about using 12-bit samples. He has some real magic with his shuffle/swing that I haven’t got my head around yet.