Thinking about trying out D&D 5 as-is (i.e. without succumbing to the urge to massively house rule it before I've even played it once!), but I can't stomach the generic modern fantasy vibe. Here are some explanations which I find more palatable for all those races.
Dwarves: men from the Iron Planet who descend to Earth to trade metals.
Dragonborn: slaves spawned in the vats of wizards. All are male.
Tieflings: victims of the shadow plague, their appearance becomes more inhuman as the disease progresses. Eventually disappear into shadow and smoke.
Halflings: gibbering semi-sentients which stalk the wastes in search of living prey to sacrifice to their idols. Among nobles of the City, it is the height of fashion to rear a captive halfling as a pet, teaching it to mimic civilised, human behaviours.
Gnomes: space pirates from the asteroid belt. Sometimes fall to Earth in meteor storms.
Elves: the construction of homunculi was once the prime mode of magical endeavour in the City. Elves are homunculi gone rogue, evolved into beings of human stature over centuries spent lurking in the shadows.
Half-elves: elves can only reproduce with descendents of their creator. Who would mate with such a being?
Half-orcs: men whose souls have been consumed by the ravenous spirits of the wastelands (known as orcs). Their bodies warped and bestial, their minds torn between humanity and depravity.