The Millennium Falcon gets its speed from the jet engine at the back, but when Han Solo drives out of lightspeed, he gets out of it really quickly. However, in space without atmosphere, it doesn't seem to make sense that it just stops instantly without anything slowing it down. How do they slow it down?
Because they're not actually travelling at light speed.
They never actually accelerate to the speed of light in Star Wars, they instead enter a "hyperspace dimension" where they happen to travel through at faster than light speeds. As a commenter noted, there was an evolution through the movies and the SW universe, so there is some adjustment needed depending on which movie you're looking at. There are a myriad of styles of hyperspace in popular culture, but here are the three most commonly known
Hyperspace is a dimension in Star Wars has to take into effect the mass shadows of stars and planetary bodies. You can finish a hyperspace jump inside a planet (although it's not a good idea, as we seen in TFA), but you have to be outside the gravity well before you can jump into hyperspace again Because of this, there are known routes that are travelled between known locations, and there is very little deviation from those routes. You're moving at the same speed, but the amount of distance covered is massive. This is most akin to a seafaring vessel sliding into gulf stream, and being pulled in a predefined direction faster than a ship could normally travel by itself... as soon as you exit the stream, you slow down to your normal boat speed.
Warp speed from Star Trek, where they warp spacetime around the ship to accelerate the ship. Because space itself is being warped, they really only have to worry about running into things when they leave the warp bubble. To use the boat analogy again, this is similar to a hydrofoil or ektoplane, where you're travelling through the water, but not really fully in the water.
FTL travel from Battlestar Galactica (new) was actually not FTL, but rather folding space... moving the entire ship and a chunk of the surrounding area to a physical location instantaneously. They just call it FTL because it IS faster than light, but it isn't actually accelerating faster than light. You can see the pros and cons of this when you see a ship do a jump inside a larger ship (taking a chunk of the other ship with them) and later when some Raptors do a jump inside the atmosphere of a planet (and one miscalculates, jumps inside a mountain). This one doesn't have a seafaring analog, but is also the same style of travel in Battlefield: Earth (the book).
Based on this article on Hyperdrives, there isn't a canon answer.
Upon exiting hyperspace an unknown technology was used to decelerate the starship.
But based on this article, ships equipped with sublight drives have other means to actually maneuver, similar to a modern jet engine craft.
Also, from the first article
To enter hyperspace, the starship's pilot would enter commands via paralight system, a combination of mechanical and electro-optical subsystems that translated the commands into a set of corresponding reactions within the hyperdrive power plants.
Based on this information, it probably boils down to the flight computer being knows the correct reactions needed in order to provide the deceleration necessary to give the full stop effect when leaving hyperspace.