The question is pretty straightforward. Is there any explanation to why DeLorean leaves burning trails in the time it travels from? Moreover, in the end of the first movie when Marty travels from 1955 to 1985, the burning trails visibly repeat his trajectory in 1985 (he moves straight, a bit to the right, then to the left, and hits the cinema doors). Is there any temporary connection between the two times?
What if the in-universe and out-of-universe reasons are exactly the same?
Because it looks cool.
In-universe, the Doc says explicitly:
The way I see it, if you're gonna build a time machine into a car, why not do it with some style?
Out-of-universe, this is a classic example of the Rule of Cool (warning: TVTropes link!).
Simple yet entirely plausible.
I'm pretty sure this has never been given an official explanation in any official BTTF product (certainly not the movies, but also not in the animated series or the video games), so that just leaves us with the intentions of the people involved in coming up with the idea and producing the effects. The book Back to the Future: The Ultimate Visual History shows the original idea was somewhat different than the finished product — there's some ILM concept art on pages 102–103 showing their idea for the sequence of visuals as the DeLorean approached 88 mph and disappeared, and each drawing in the sequence has a caption; they read as follows:
- Electrical EFX from headlights, racing to rear of car.
- Front end heats up, starting to form fireball.
- Car now totally engulfed in flame; long tail forming.
- Fireball turns to meteor; mass of car thins out.
- Front of car reaches time barrier; streak zips into opening.
- Streak follows car; leaving behind fire trails where tires were, fire trails then zip into opening along with papers.
Both the drawings and the text make clear that the fire trails were at that point supposed to be in front of the 'opening' in spacetime the DeLorean had passed through, i.e. where its tires had actually been driving right before it disappeared, and that the trails were a remnant of the fact that the DeLorean was supposed to be engulfed in flame as it approached 88 mph. (Not sure where the "papers" in step 6 were supposed to have come from, perhaps just trash in the parking lot pulled in by some kind of suction towards the 'opening'.)
Unfortunately, I can't find any information on when this was changed so that the fire trails extend past the actual point the DeLorean disappeared, or who came up with the idea. It's not discussed in the above book, or in the DVD commentary transcribed here, or in the book We Don't Need Roads: The Making of the Back to the Future Trilogy. We Don't Need Roads does seem to say on pp. 67–68 that the basic idea of the DeLorean being fiery as it left was from the illustrator who came up with the design for the DeLorean time machine, Ron Cobb:
While working on his round of design sketches, Ron Cobb mentioned that he believed the time machine would go out through the time portal at a burning-hot temperature, but come back cold. To that end, the special effects team lit fire trails whenever the DeLorean was beginning a journey.
Cobb also gives more information about his reasons for want the DeLorean to "come back cold" in this interview (it has to do with the idea that the DeLorean briefly passes through some kind of hyperspace dimension which is at absolute zero), but doesn't talk about the fire trails.
To speculate, the motive for changing the fire trails from being where the DeLorean had actually driven to making them go past the point that it disappeared may have just been to have the cool visual of Marty and Doc standing on the fire trails, which wouldn't have made sense if the fire trails only went over the parts of the ground the DeLorean had actually been driving moments before. And once you show the fire trails going past where it was actually seen driving in the departure time, it seems like a logical extrapolation to imagine the fire trails appear along the same path the DeLorean will take immediately on arrival in the destination time.