Jump drives are point to point hyperspatial engines. Jumping through hyperspace is fast and dangerous, though the affini handle it a lot more safely. Older terran ships had a drive engineer to manually plot a jump, while the ships made in the few decades before encountering the Affini Compact had managed to automate the process.
So the simple bit is that a terran jump drive punches a big whole into hyperspace and back out at another location and moves through it in an instant. There's usually a flash of light and a ship just sort of vanishes into hyperspace and back out a moment later into a new location.
Nobody in the Terran Accord actually understood how it worked. They had theories and models that covered some of the behavior, but since they didn't discover the 5th fundamental force there was a lot that wasn't known. Because of that ships sometimes got lost and no one ever knew why.
Hyperspace jumps cause most sophont species(except affini!) to feel a hypermetric kick. When you make holes in space time it sends out ripple in the fabric of the universe. Creatures with hollow cavities in them(like terran lungs) can feel the ripple. Terrans call it a hypermetric kick because it feels like getting kicked in the lungs. It doesn't actually apply force, it's only a sensation.
You don't want the public explanation you want that hard science. Well Hyperspace is a chaotic mess of undefined reality that exists beneath spacetime. There are special kinds of matter called exotic matter that when you make it a certain temperature it interacts with space time making a hole into hyperspace.
Now getting into hyperspace with exotic matter and some powerful electromagnets is easy—relatively speaking—getting out is the hard part. Getting out where you want to go is an even harder part. Something you'll need a degree for, maybe more than one.
To put it in 101 terms, a powerful precision electromagnet lets you aim, and the fuel(exotic matter) lets you punch a hole into and out of hyperspace, and a specially designed composite hull keeps your ship from being instantly destroyed when you do that.
Of course that generates a massive amount of heat and fills your engine with sublimated exotic matter which becomes regular radioactive matter at that point. So that means you have to wait for the drive to cool down and flush out the used up fuel. Most of the fuel is converted into energy but the remains get left in the void. In space radiating heat is extremely difficult since conduction and convection are out, so we rely directly on radiating heat to the void. That takes time.
So what happens when your captain tells you to jump gain immediately? Bad things. Tell them to fuck off, politely. So you just jumped and you have a super-heated jump drive trying to radiate immense amounts of heat around everyone's living quarters. You've got an entire containment unit for unused fuel because if it gets above -100 C that unused fuel is now a hypermetric explosion. Do you open up the valve to funnel highly dangerous and temperature sensitive exotic matter into a super-heated drive chamber? No. It'll react before the drive spools up and it'll smear your ship across hyperspace. Alternatively, it works! And you just generated enough heat to cook the entire crew alive because the ship was already overheated and you added a fuck ton more… and then that also breaks the containment unit and the hypermetric explosion turns into your cremation. Oh and you just blew a dangerously large hole in space time for everyone else who has to hope they aren't traveling near it. Again don't jump before you cool the drive.
So that's in the most basic terms, the terran theory and mechanics of jump drives. Its dangerous, inefficient, and violently damages the fabric of spacetime assuming it'll just fix itself.
Okay little one, I'll teach you a bit about our "jump drives" but please remember to eat, drink, and take your meds first.
Where your adorable little jump drives poked holes in space ours glide and soar through the fabric gently. The traction engine on this ship is hyperspacial—pieces of it are physically in hyperspace. Those same pieces root themselves in place to gain traction and propel our ship.
To enter we use much smaller amounts of exotic matter to weaken the fabric and then the traction engine pulls us under and maneuvers us through hyperspace and once it is in position it will reverse the process, weakening the fabric again and pulling the ship back into spacetime.
It only takes a moment or two same as your terran ships but the entry is far gentler on both the ship and spacetime itself. It is also accurate—not like a terran one where an accurate jump lands you in orbit of the right planet—accurate to less than a nanometer. How do you think we jumped right on top of your ship, our vines held your vessel before you even knew we were there.
Since we use different kinds of exotic matter and in smaller amounts, there is no waste product, and the cooldown times on affini jump drives are minuscule compared to terran ones.
Oh you're wondering why your little vessel couldn't jump away? It's very simple. Some forms of exotic matter weaken or punch holes in spacetime, other strengthen it. Of course that means no ship (affini or terran) can enter hyperspace in the area, and any ship that attempts to exit hyperspace there well… let's just say the ship doesn't survive. Affini ships can detect and navigate around a suppressed exit point but terran ones can't.
Yes dear it's as simple as that. A simple tool that we have to be careful about applying when catching little fishes like you.