For centuries, it had been believed that no object with mass could surpass the speed of light; even today, that rule is insurmountable, if only by technicality. However, the introduction of eezo allowed for a loophole in the laws of the universe to be exploited, if nothing could go faster than light, then make light go faster.
Modern starships, through the use of their eezo cores, accelerate the speed of photons near the ship's hull, increasing the speed of light and allowing for a vessel to travel faster than the speed of light in a vacuum, while not surpassing the speed of light around itself.
When moving under the effects of an eezo core's mass effect field, a vessel is still moved by its sublight engines, but with far greater speeds easily achievable. The very difference in the speed of light completely changes the speed a ship's engine can propel a vessel through scientific laws still not entirely understood by first contact.
There are still limits: a vessel traveling in FTL expends fuel at a rate more in line with the rate normal for the vessel traveling without the lowered mass brought by eezo, greatly increasing fuel consumption compared to STL travel and making FTL uneconomical for most moderate range intrasystem travel.
The dispersal of the accelerated effect on the photons is not easily predicted, with countless factors, many unknown to mankind at the time of first contact, affecting the speed of dispersal, and thus exit time from FTL. While the Citadel Races have managed to discover most of the variables and account for them, the extreme amount of variables needing to be accounted for, from vacuum molecular density, to vacuum eezo density, to gravity fields, to even the amount of FTL use in an area and the few variables that not even the most advanced races can reliably trace without pure trial and error, such as the Dark Matter density of an area, makes FTL accuracy in even the most documented areas imprecise at best.
Travel through FTL leads to static buildup in the eezo core of a vessel. If left unchecked, the charge will lash out at the craft and do horrendous damage, killing the crew and crippling the vessel. An eezo core can usually contain the equivalent of one hour of FTL static buildup at a core's maximum output (not connected to the speed of a vessel, the same core on different ships will produce different speeds at max output) per tenth of a percent of a ship's mass the core's eezo represents in the most basic drives. Alterations in the micro construction of an eezo core can allow for an increase in capacity of up to 50 percent in the most advanced models fielded by the Asari, but humanity and the vast majority of galactic civilians lack such technology, with only military grade vessels and the most expensive civilian craft being equipped with such drives (and even those upgrades seen upon most craft only reaching a 20 to 40 percent upgrade).
A static charge can be dissipated into another object, though said object needs to be significantly larger than the craft discharging into it. Larger vessels that can't enter a sufficiently large object's gravity must discharge into a planet with a strong magnetic field, typically as a gas giant. This requirement for a large gas giant limits the systems a large vessel can operate within effectively, leaving many systems solely serviced by Schooners, smaller Cruisers, and specialized (and relatively expensive) Cargo Ships, for their transport needs. Schooners in particular gain their prevalence due to almost ignoring size restrictions for discharge points, their small size allowing them to use even dwarf planets as discharge points if needed.
The greatest limiting factor, however, is the Gemini Limit. Named for the test ship that learned this rule the hard way. The Gemini Limit creates an effective cap on a vessel's FTL speed due to matter in the vacuum of space. In most situations, the discordance between a FTL bubble's speed of light and the true speed of light will bend matter in space around the bubble, the material taking even obscene detours to avoid the discordance. However, as a FTL bubble increases in velocity, and the volume and mass of matter it displaces increases, it begins to strain, eventually "popping". While the popping itself does no damage to a ship (the vessel just immediately going from FTL to its comparable percentile in normal C), such an event can make an eezo core explode, ripping a ship the pieces via mass discordance and shifting gravitic fields, as happened to the Gemini. The speed cap depends on the density of matter in an area, speeds are significantly limited inside a solar system when compared to interstellar space, with most ships only able to go a few times the speed of light within a system.
The speed a drive can take is based on the volume the bubble takes up, and a drive's power. This gives two paths for increasing speed, getting a bigger eezo core, or making a more efficient, skin-tight bubble.
Increasing a core's size has three main advantages: it allows for more power to go to the bubble without making the eezo explode, increasing a ship's speed, it allows a ships to become lighter when outside of FTL using the additional eezo, and it allows a drive to absorb more static electricity before needing to discharge. The main downsides are that the core requires more power to operate, and a bigger core gives diminishing returns. While increasing a core's size works well until around 6 light years per day, any further and you need exponentially more eezo to gain higher speeds, at a level it is entirely uneconomical both from a cost (important for civilians) and expended mass (important to the military) perspective.
More efficient bubbles can be created through the use of mass effect field shapers. Shapers are what allows for any complex use of mass effect fields like artificial gravity, micro material manipulation, kinetic barriers, or inertial dampening. Mass Effect Fields are naturally sphericall, but the protheans (most likely through observation of biotics and their natural field shaping capabilities) discovered a way to alter a mass effect field into different shapes, greatly increasing the versatility of them. (This is one of the reasons the discovery of prothean ruins are so significant to a developing race, while they could be using eezo, their use of it could only be extremely simple without the ability to shape fields. It's entirely how eezo, an element with only mass altering properties, can have gravitic effects, a hyperdense field is placed very near an object, effectively creating an area with similar gravitic properties to a black hole without the radiation and other negative properties of one.
Shapers used to manipulate a FTL bubble are not very precise, Citadel tech not being able to manufacture shapers with the precision the protheans had and the manipulation of FTL bubbles being notoriously difficult. The shapers are still able to change the spherical shape of the bubble into something more in line with a vessel's shape. The more precise the shaping is, the more volume efficient a drive is, and the faster it can go. Most civilian grade shapers can get a 20 to 30 percent increase in speed over the standard for a drive to mass ratio. Military shapers can get 40 to even 50 percent increases, with the latter only on the most advanced systems, usually not used on standard warships due to their prohibitive cost.
Human vessels before first contact had relatively limited shapers, with earlier vessels before late 2150 almost all lacking them. Those shapers in use are less effective than most in Citadel Space, with them providing only around a 20 percent speed improvement, with military units not any better.
The shapers of a ship can be more easily replaced than its eezo core, but are also easier to damage (being on or near the outside of a ship) and still being relatively expensive pieces of equipment, making their replacement costly, though not as much as getting a new eezo core.
The vast majority of sophisticated field shaper production, including the majority of military grade production, reside within the Asari Republics, specifically the republics of Armali, Siruna, Dalivrun, and Te'sarun. Major military powers attempt to ensure enough military grade domestic production exists to support their own militaries, but many minor and even intermediate powers like the Hegemony are reliant on Asari imports to equip their warships.
Naotw: Thanks for catching that. Most of the information is just giving reasons for things that already exist in cannon (how it actually works), reiterating cannon (static charge), or explaining common fanon (FTL speeds, though ME itself began drifting from the ideas in the FTL codex pretty early on with giving actual speeds instead of acceleration-deceleration). The added info won't really change much.
There are several stars within that area of Sol, and much of the galaxy is far denser than the Local Cluster, I've always found smaller Clusters to be logical due to this, alongside the close proximity of Arcturus.