Pieces like this are about your understanding of rudiments and your ability to play them, so there are a few things you should know:

Part 1 is about your ability to switch between open and closed rolls, as noted (duh). Open = double-stroke, closed = buzz.

Part 2 is testing your ratamacues (again, duh). The sticking on these should alternate: llRLRL rrLRLR

Part 3 throws a bunch of rudiments together. Single grace notes are flams: lR rL. Double grace notes are drags: llR rrL. Triple grace notes at the end of the piece don't have a name (that I know of) but should be played: lrlR rlrL.

Three slashes above a note is a sort of shorthand for 32nd notes, or for percussionists, double-stroke rolls. Buzz rolls are marked with a Z or are called out as such: "closed" or "buzz" or "bounce"

Check out the "40 Essential Snare Drum Rudiments" in the sidebar, and take some time to write out the sticking for the piece, making sure the sticking for the main notes alternates. Then go back and put in the appropriate sticking for the grace notes.

E.g. Measure 10 of Part 3:

RLR LR LR LRL (main notes)

lRLR rLR rLR rLRL (with grace notes)

The exception here is with the ties on the rolls and the stroke after the roll. I.e. think of the sticking on tied notes as one note for sticking purposes, not two. You start the roll R and the first note after it is also R; the next note is L.

Lastly, the four quarter-note double-stroke rolls in Measure 9 is a little bit tricky. It should sound almost like a whole-note roll, but with an ever-so-subtle accent on each downbeat. Each will start with R, and not alternate.

Again, this is a test of your ability to play rudiments, which is essentially about sticking.

The other answer here is not entirely technically correct, and when it comes to rudiments, technically correct is the only kind of correct. :)

Good luck!