It's a common thread throughout the vampire genre. According to Bram Stoker, you cannot see a vampire's reflection. Where did Stoker get the idea that a vampire's reflection cannot be seen?
It is said that when you are looking into a mirror you are looking into your soul. So without a soul (as it is commonly accepted that vampires do not have souls), there's no reflection. Mirrors also used to be made with silver (not entirely of course, because whoa, expensive, but it was used with the other reflecty material). Silver is said to ward of evil and is bad-news for the undead (vampires included) so that probably would have added to the whole "mirror + no soul = great way to vampires" thing.
Mirrors were also said to ward off evil and since vampires are commonly associated with a lack of soul (therefore evil by superstitious folks) that could also have something to do with it.
Also, if I recall correctly, being unable to see a vampire's reflection was made popular by Bram Stoker's Dracula. So that took hold and bam! Vampires without a reflection = common theme in the Vampire genre.
In the Doctor Who series 5 episode The Vampires of Venice, yes, there is an explanation. In the Doctor's own words:
You're using a perception filter. It doesn't change your features, but manipulates the brainwaves of the person looking at you. But seeing one of you for the first time in, say, a mirror, the brain doesn't know what to fill the gap with, so leaves it blank ... hence no reflection.
This is, of course, what one might call a "sciency-whiency" explanation, to match the science fiction genre, of what is usually a phenomenon found in fantasy literature. However, it is what you asked for: an explanation of why vampires' reflections cannot be seen, at least in one work of SF/F.
I was asked to copy my answer from mythology.
In a nutshell: While it's still probable that Bram Stoker did not think of the "no reflection" feature, it's considered his invention.
Back in the past, mirrors were made out of silver, and actually Balkan werewolves are considered a type of a vampire, so... if you add 2+2 u get 4. It's quite possible that the modern myth is a combination of the silver and soul reflection theories.
A full detailed explanation and references can be found in the full answer on Mythology.
There could be multiple reasons for a vampire's reflection not to appear. Having no soul is one theory, but in another theory, it is simply an adaptation to make sneaking up on victims easier. Regardless of the reason, not having a reflection is certainly helpful to the vampire.
The other answers are mostly true or pieces of the truth.
When a person died and then came back from being burried (due to them not actually being dead) it was considered that they were "possessed" or their soul "corrupted". Left alone they would eventually become Hell hounds/Demon Dogs which is why they're associated with Werewolves.
Silver is considered purifying and "clean" substance. I do not remember the origin of why this is, but it might have to do with it being an effective cleaning agent in reality.
Being that Vampires and Werewolves are "corrupt/impure" and Mirrors were made of silver and were sources of purification, a corrupted soul would not reflect. This is likewise why Silver works in killing werewolves, it rejects and disperses the corrupted soul of the werewolf.
In other words. Mirrors reflect truth and Vampires have no soul or are evil thus the truth is that the person that is the vampire isn't really there, thus would not be reflected OR would be reflected as monstrous.
Stoker's Vampire did not reflect and became popular so most sources forgot that not all Vampires don't not reflect when they used Stoker's story as a source.