SYSTEMATIC= Having, showing, or involving a system, method, or plan.
We can't say that a car is "systematic". They only use this word here because it ends in "-atic" and rhymes with the following ones, but none of them can be applied to a car, they simply sound complicated and sophisticated, like this car.

HYDROMATIC= This word doesn't exist. (see "systematic" above), but it makes us think of water (hydro) and automatic.

ULTRAMATIC= Packard's Ultramatic transmission was the creation of the company's chief engineer Forest McFarland and his engineering team. The magnitude of the accomplishment is illustrated by the fact that it was the only automatic transmission produced solely by an independent automaker with no outside help.

WHY= An expression of surprise or shock similar to "oh yea" or simply "oh".

GREASED LIGHTNING= Lightning is the visible flash of light accompanying a thunder in an electric storm (see picture). If you say that a car is lightning, you mean that it is really really fast (fast as lightning). GREASED is something that has been covered in grease. If you grease a car engine with car oil it will work better and the car will go faster, so a greased car goes better and faster (see pictures of grease). The name of this movie is GREASE, but meaning a kind of greasy shiny gel used to sculpt your hair and make it stiff and wet-looking (like John Travolta's hairstyle in this movie), and this song uses the word "grease" with the more normal meaning of greasy oil.

OVERHEAD= Something overhead is higher than your head.

LIFTERS= A piece of machinery used to lift things or people up. Also, in a gym, a lifter is a machinery used to lift weight up by pulling down and develops your triceps and shoulders.

QUADS= (informal or slang word used for some things with 4 parts) Quadriceps muscles; Term referring to a 4x4 dirt ATV (all terrain vehicle). Gun with 4 barrels (the tube shape part of a gun). I supposed that "4-barrel quads" here may refer to some kind of sophisticated car engine with 4 camshafts, making it especially powerful.

FUEL INJECTOR= A device in the engine that injects fuel into the combustion chamber, making it faster and more powerful.

CUTOFFS= Shorts made by cutting the legs off a pair of trousers, especially jeans, especially when used by girls (see picture).
The song is simply mixing vocabulary from cars with some kind of sexy innuendo. The corresponding car term is "a deceleration fuel cut off", which is the opposite of a fuel injector, it suddenly stops the injection of fuel into the combustion chamber when the vehicle suddenly decelerates in the event of a crash, preventing the engine from exploding.

CHROME PLATED= Covered with a thin layer of chrome. Chrome is a paint containing chromium, and chromed objects are shiny and look similar to silver. It is also used to prevent the iron from rusting in the engine.

RODS= Connecting rods (a part of a car engine linking the piston and the craftshank.

I'LL SEE YOU GET THE MONEY= I will make sure that you get the money.

A FOUR-SPEED= A manual transmission with 4 forward gears in either a car or a motorcycle. So here, "a four-speed" probably means simply "a car" (having four speeds). FOUR ON THE FLOOR= A four-speed manual transmission with floor mounted shift. In the song, "a four on the floor" is a car having that technology.

THEY'LL BE WAITING AT THE DOOR= There will be many people wanting to come.
If a place (like a disco) is very popular, it quickly gets full and you have many people waiting outside, at the door, trying to get in.
In this case, "they" are girls. With this wonderful car, many many girls will want to have a ride with you.

THAT AIN'T NO S*T= A rude and very colloquial expression with a terrible grammar (a double negative and a swear word) meaning: I'm serious, I mean it, I'm not joking, it's true.

WE'LL BE GETTING LOTS OF T*T= A rude and sexist way of saying "we will have a lot of girls".

IN GREASED LIGHTINING= With this wonderful superfast car.

GO, GO, GO...= Come on, do it! (an expression of encouragement).

BURNING UP THE QUARTER MILE= Going very very fast along the "quarter mile". That distance (1/4 mile) was the most common one in street racing.

COASTING= Moving effortlessly and smoothly.

THE HEAT LAP TRIALS= The one-lap race or "heat" to determine starting positions in the actual race.

SUPREME= Wonderful, extraordinary, the best.

CHICKS= (coll.) Girls.

CREAM= (slang) Too rude to explain but if the girls will cream for this car, it means that they will love it.

FRENCHED TAILLIGHTS= Taillights with a sophisticated design that makes the car look really cool. TAILLIGHTS (AmE)= Rear lights (BrE): the red lights at the back of the car.

FINS= Wings and airfoils used to improve a vehicle's aerodynamics, stability, or possibly aesthetics (see picture).

PALOMINO DASHBOARD= A palomino is a horse with a white mane and light brown fur (see picture). Palomino leather is leather in that colour (see picture). Palomino leather interiors were the top of 50s and 60s automotive fashion, sometimes including palomino leather dashboards. A DASHBOARD is the part of the car where the wheel and all the indicator dials and switches are placed (see picture).

DUAL MUFFLER TWINS= Two exhausting pipes at the back of the car (see picture)

PISTONS= Parts of the engine that turn the explosion power into movement for the wheels.

I CAN GET OFF MY ROCKS= This sentence can mean different things, but it has a strong sexual meaning we're not going to talk about. It also means to have a very good time.

I AIN'T= (slang) I'm not.

BRAGGING= Boasting. If you brag or boast about something, you talk about it in an arrogant way trying to prove that you are superior because of it.

PUSSY WAGON= Another sentence with sexual innuendo.
Many boys use "she" when talking about their car. WAGON is a slang term for car. PUSSY is a female cat and some other things, and it may be used to talk about a hot girl (a sexist term). So if the car is a "she" and she is wonderful, you can say that "she's a real pussy wagon" (= it is really a sexy car).