On the Road Again
In the beginning, there were jumpgates: pairs of portals linked by an Einstein-Rosen Bridge (wormhole) which could allow near-instant transit between two distant points in spacetime, for the low-low price of a few kg of antimatter per jump. Then, the Earthgate Drones came, and the gates became a pathway for invasion: as a result, they have all been destroyed.
So how does anyone get around?
The gates were a great way to travel fast, but not the only way. The Abelites perfected a practical Alcubierre Drive in the 24th century; the drive allows travel by folding spacetime, but is prohibited by the laws of physics from allowing superluminal travel. A particularly-efficient medium-size ship might have a warp flight time measured in days, or even weeks for special long-range craft; at sublight speed, this drive is not sufficient for interstellar travel.
In the early days of the Homeworlds, long-haul sublight probes would embark on decades-long missions to deliver a jumpgate to a habitable system. Once there, the gates could allow virtually instantaneous travel. Thus, the Homeworlds' territory expanded in the early centuries in a rough sphere, growing at high sublight speed. Today, there exist colonies over 600 ly from any of the Homeworlds, although the vast majority of DMZ citizens live within a 100 ly sphere.
Even so, 100 light years is a daunting distance at sublight speed. The people of the Homeworlds got used to being able to jump between far-flung systems instantly, and after Earthgate, there needed to be an alternative FTL system; otherwise, humanity would fragment, and most colonials were doomed.
Once again, the Abelites saved the day. For centuries, they had been working on the technology to create wormholes without jumpgates, and they had proven the concept well enough. Until Earthgate, there was no particular reason to develop the technology, as jumpgates were so reliable and efficient. Earthgate kickstarted the Jump Drive, and within a year there were ships jumping from the Homeworlds to the nearer colonies to determine their fate and offer relief.
The Jump Drive
The Jump Drive, as it is colloquially known, operates on a fairly simple principle. Imagine spacetime using the classic analogy of the two-dimensional rubber sheet, with gravity wells forming dimples into the third dimension. A ship descends into a substantial gravity well--say, that of a sun. It is then possible to "aim" a "straight line" toward another gravity well; assuming there are no other suns or black holes between the two, the path is clear. The two wells can be briefly connected by a wormhole, "tunneled" by the Jump Drive and automatically creating an exit mouth where the "straight line" intersects the destination gravity well.
This analogy is hilariously inaccurate, but actually does a good job of modeling the most important features and limitations of the technology. The most crucial element is a significant differential in spacetime curvature between the endpoints and the path. In the analogy, a heavy star in the path of the "tunnel" would block travel; in actual use, this holds up. Spacetime can curve for many reasons, so jumping between two points of relatively flat spacetime is more or less impossible--you will end up at some random point along the "path". Jumping from deep well to deep well minimizes the possibility of getting lost.
The Jump Drive's greatest limitation is also, by happy coincidence, the source of its relative efficiency. Creating a "tunnel" many light-years long requires a staggering amount of energy. Even with antimatter as an energy storage device, carrying all this energy on board would be prohibitive. But as long as you're closely orbiting a star, you might as well use all that free energy.
Most modern jump-capable ships employ solar collectors, sometimes kilometers in diameter; at close orbit, they can capture terawatts of solar power, enough to charge the massive capacitors needed for a jump in a matter of minutes or hours. This process is much more efficient than generating power locally, or carrying stored energy (such as antimatter), so for most purposes, this technique is used.
Jump Gates
A Jump Drive need not transport the ship that carries it. Indeed, a new generation of "Jump Gates" has sprung up, installations in permanent close solar orbit which create wormholes on a regular schedule. Ships up to high-subcap size will use warp drives to approach the gate, enter a queue, and find themselves guided into the tunnel mouth and transported to a far-flung star within hours or even minutes. These tremendously-useful installations are some of the grandest spaceborn constructions ever built by the Homeworlds, and are confined to densely-populated systems.
By accord, all Jump Gates are secured by the Foundation, even in Imperial space (although in the latter case, the duty is often shared). This is something to keep in mind, as individuals with a high Security Standing may not be allowed to use the gate, or may even be apprehended on approach. On the other hand, it also means that individuals with low standing with an Empire may travel through their space, so long as their Foundation standing is solid, provided they do not leave the DMZ which typically extends one million kilometers from the gate.
The Abelite Imperium graciously donated all knowledge on the construction of Jump Gates to the Foundation, the better to allow all to benefit from this technology, without regard to imperial politics. That being said, the gates are built on centuries of Abelite industrial know-how, and virtually all important components of the devices are manufactured by Abelite firms (or NTC firms operating under strict secrecy agreements). The other empires are hard at work, and will likely catch up in a matter of decades, but the fact remains that, were a major imperial war to break out today, the Imperium would enjoy a tremendous advantage, possibly having back-door access, or knowledge of certain tactical considerations that need not have been shared for civilian use of the gates.
Jump Ships
To reach more remote destinations, one needs one's own Jump Drive. Since the mechanics of wormhole creation require exotic hardware--both to produce planetary-scale power and to harness it to create exotic local spacetime conditions--no ship smaller than a capital ship can efficiently mount a Jump Drive. Military capital ships, and the largest industrial transports, typically have their own drive, while others will tag along with a Jump Ship: a massive capital carrier, mostly empty, with room for many sub-capital ships within its hold. Such ships will make runs where economically-feasible; that is, without a mostly full hold, the trip isn't worth the cost, so the more remote the destination, the more infrequently the route is flown. This is compounded by security concerns; the value of a Jump Ship, regardless of contents, is worth the entire fleet of the grandest of pirates, and many bands would stop at nothing to hijack one. Jump Ships won't jump into low-sec systems without escort, requiring the assistance of other militarized capital ships--a very expensive proposition.
The Foundation does fly a number of armed Jump Carriers specifically for the purpose of connecting the remotest colonies to the major FTL hubs. Having had only two decades to ramp up operations, they've only managed to outfit a small number of these ships, so some sparsely-populated colonies in particularly low-sec space will go months between jumps.
Foundation Jump Carriers operate with discounted fares, and frequently carry over capacity, offering free transport to anyone in need for humanitarian reasons. These ships are often packed to the brim, in both directions, and are a constant subject for political discourse. Some favor ending the program, arguing that the forcible relocation of just 1/10th of one percent of the Homeworlds population would remove over one third of the colonies from the FTL network, cutting the costs of free and discounted Jump Carry services by upwards of 80%. It is a powerful argument, but the ethical considerations have tied up the proposal for years.