I can’t find any suggestion that the giants had anything to do with the origin of the elves; the elves seem to predate the giants’ rise to power, actually existing during the Age of Demons, at which time the “lesser races” (i.e. not demon or dragon, apparently including at least the elves, giants, goblins, and orcs) simply tried to avoid getting killed in the war between demons and dragons.
After the couatls sacrificed themselves to end that Age, the giants were able to rise to power in Xen’drik using dragon-taught magic, ushering the Age of Giants.
The giant empire then enslaved elves, and thus “inadvertently” raised them up out of their primitive state, teaching them magic and the like. Towards the end of, or just after, the war with the quori, the elves revolted. The giants were going to put this down using the same cataclysmic magic they’d used against the quori, but the dragons came and crushed them, instead. An elven prophet/hero named Aeren, among other things, foresaw this end and signaled that the elves should flee Xen’drik before it happened.
Those elves named the island they fled to “Aeren’s Rest” or Aerenal, and referred to themselves as the Aereni. Aeren had sacrificed himself during the exodus, but not before figuring out how to do the whole undying thing, so he became the first undying councilor.
The drow, of course, stayed in Xen’drik. Simply based on the fact that all elves from Xen’drik are drow and no Aerenal elves are drow, it seems likely that the actual distinction between “elf” and “drow” happened after the exodus, probably during or immediately after during the dragonfire-purge.
However, Secrets of Xen’drik does have this:
According to one legend, the Sul’at giants created the drow by binding dark forces to normal elves
The Sul’at League was a series of fire giant city-states, founded by the first fire giant Adaxus, who got that way through some fiendish pact. Or so legend has it, anyway. This is the only mention I can find of giants manipulating an elven race. That said, it doesn’t seem to jive with the facts, which calls that legend into doubt. It seems unlikely that every single drow would choose to stay and every single elf would choose to leave.
The elven trance is different from sleep, and their “dreams” are different as well. I am still searching for information about whether or not these dreams are similar enough that they still go to Dal’Quor in spirit while they trance.
Unfortunately, I have so far looked through Eberron Campaign Setting, Secrets of Xen’drik, Secrets of Sarlona, Magic of Eberron, Races of Eberron, and Player’s Guide to Eberron, all without finding any reference to elven dreaming or lack thereof, much less whether or not the giants were responsible for it. I didn’t read those books cover-to-cover, but anything that sounded like it might even possibly have the reference, I checked. I’m running out of even remotely likely suspects.
An elf receives the benefits of a long rest in 4 hours while using the "Trance" trait.
According to the 2017 update to the Sage Advice Compendium:
Q: Does the Trance trait allow an elf to finish a long rest in 4
hours?
A: If an elf meditates during a long rest (as described in the
Trance trait), the elf finishes the rest after only 4 hours. A
meditating elf otherwise follows all the rules for a long rest; only
the duration is changed. This answer has been altered as a result of
a tweak to the rules for a long rest, which appears in newer printings
of the Player’s Handbook.
This ruling reverses guidance in the earlier version of the SAC, due to errata changing the rules for long rests.
Interactions between the "Trance" trait and long rests
A long rest is defined as:
... a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours and performs no more than 2 hours of light activity, such as reading, talking, eating, or standing watch.
The elf's Trance trait is defined as:
Elves don’t need to sleep. Instead, they meditate deeply, remaining semiconscious, for 4 hours a day. (The Common word for such meditation is “trance.”) While meditating, you can dream after a fashion; such dreams are actually mental exercises that have become reflexive through years of practice. After resting in this way, you gain the same benefit that a human does from 8 hours of sleep.
Since the "Trance" trait replaces the need for sleep (which most races need in order to complete a long rest), the elf is able to satisfy the requirements of the long rest while in a semiconscious trance for four hours.
Best Answer
It doesn't mean anything special in D&D, but is rather just an English expression.
In this context, "dream after a fashion" is an idiom meaning that you can "kind of" dream. It's not real dreaming the way other races experience dreams, rather it's something elves can do that mimics dreaming. This is why it's followed by the phrasing "such dreams are actually mental exercises" - they're not dreams the way we think of them, rather the elvish version of dreaming.
As described, they basically do a kind of meditating via mental exercises that might look like dreaming to an outside observer. The elf in question goes quiet and still, as noted by the use of "semiconscious", and is likely unresponsive to outside stimulus until the four hours are up.