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In the player's handbook there's a spell called Dimension Door. I know it is used for teleportation but everything else is vague.
How does it really work?
Does the user have to use the door to teleport? If not, then why is it called "Dimension door?"
No, you don't have to step through a door (magical or otherwise) to teleport. Dimension Door only has a Verbal component, and its duration is Instantaneous. So you say some magical words, and you're instantly teleported. The description doesn't say anything about a door; it's only the title that contains the word "door".
As for why the spell is called Dimension Door, this particular spell goes all the way back to OD&D (I believe it's been in every D&D edition), so the name in 5e is simply a legacy of previous editions.
An open doorway to a different area of the battle was what the spell looked like in previous incarnations in all artwork associated. Some versions showed the doorway as a shimmering portal and some showed it as an open doorway to that showed the location the caster was teleporting to.
Although, as Miniman's answer stated, the only requirements in 5e is a verbal component and thus you don't have to 'step through' the doorway, it is still completely feasible and possible to describe this as a doorway.
As an additional note of inspiration for the name's modern interpretation, in 4th ed's case, teleportation spells took you through other dimensions in order to teleport. The Feywild was a common route, though the Shadowfell was sometimes used. This was written in to the descriptions of several racial abilities. IIRC, Dimension Door included the same description. So in 4e the emphasis was on the dimensional travel and not the door. 2nd edition and prior also related teleportation spells to extra-dimensional travel (though primarily through the Astral Plane).
After looking further I did find one reference, albeit possibly circumstantial, that alludes to the ties between teleportation and dimension travel. The details of the spell Hallow have a specific subheader about 'affected creatures' that states the following:
Extradimensional Interference. Affected Creatures can't move or travel using teleportation or by extradimensional or interplanar means.
This is not specifically stating that teleports are dimensional travel, but the name of Dimension Door and the text of this passage in this spell allude to it highly, as does the fact that spells that discuss warding against teleportation ward against extraplanar travel and vice versa (examples are Forbiddance, Forcecage, and Planar Binding).
Conclusion
While it isn't specifically stated in 5e, the name Dimension Door alludes to traveling through another dimension in order to move quickly, as does the history of the spell and references of other spells. The aesthetics of the spell are primarily up to the player and the DM collaborating, but visual references do typically use the shape of a doorway to represent the spell.
Why is it called "Dimension door?"
Actually, back in the day, you would open a small portal in front of you, take a step through it, and be at the destination. Functionally though, dimension door is any quick battlefield teleportation. For 5e they kept the traditional name and the same battlefield function, but tweaked the "special effects."
So even though there's only a verbal component, the legacy still holds in the 5e illustration of the spell:
There was a cinematic from one of the old D&D computer games that showed a couple of mages using dimension door, and it was an actual door.
This is how the spell was illustrated in 4E:
And this is how it was rendered in Baldur's Gate:
