I'll answer this in the most likely answer there can be. Someone screwed up in the we didn't think this out because it just looked cool department.
While the door/ramp looks to be on a winch(s) (the extra ramp parts have red wheels as seen at 0:18, to allow them to fully extend), unless Doc could remote control the box truck, there is no possible way for him to get in and back out the Delorean.
Movie goofs, they happen.
There is only one explanation for how he could have done it, and it's almost Rube Goldberg in its nonsensical construction, even for Doc. He would have had to move the truck, lowered the ramp, got into the back, closed the ramp, climbed into the Delorean through an open window, allow the Delorean or something else to fill the cabin and the open windowed Delorean with smoke, closed the window, then open the ramp again to pull the Delorean out, all before the smoke dissipates. Even Doc isn't that freaking pointless or convoluted (I hope).
But as @WadCheber has linked to, the Delorean window is about the size of a bread box, making even that convoluted idea impossible without magic.