On a purely psychological level, providing emotional support to someone will eventually mentally wear you out, as will being around people who are expressing negative emotions, as those emotions will tend to affect your own. Conversely, anyone who walks into a room-full of happy energetic friends usually can't help but perk up a little themselves.
Concerning the metaphysical level, there is a not uncommon belief among several religions and systems of magick that there exists an energy that arises from all living things; also, that this energy can be manipulated and exchanged in a variety of ways, and it is likely that there are at least some individuals who do so without realizing it. What follows is a very broad summary of said beliefs in the spirit of anthropological study, accumulated by talking with various people over the years; in reality the details may vary widely from person to person.
Energy exchange can take place in a number of situations and in varying volumes. Sometimes, when providing emotional support to someone, the energies of both will mingle a little, and the supporter might donate small amounts of it. This is normal and natural, especially if there is physical contact. A person under severe emotional stress may unconsciously seek to take in energy because they need it to maintain psychological/emotional/spiritual stability. Generally, if they don't realize what they're doing, they will not take more than they need, and even that amount is too small to be truly dangerous to the donor. Low energy levels get dangerous when they affect one's normal functioning, just like physical or emotional fatigue. Theoretically it should be possible for someone to die as a result of having too much of their energy drained.
Psy-vamps take more energy than they need and/or take it even when they don't need it, and there seems to be one primary reason: the taste and feel of that life energy in sufficient quantities is apparently quite intoxicating. No physical form of sustenance can hold a candle to it, the closest conglomerate description I can make is is to tell you to imagine drinking sunlight, nice calm soaking up of that energy... now imagine something ten times that potent, and taking it in ten times as fast, only now there is a beating heart behind it, and it's so alive and full of colors and nuances and flavors that you can't get from sunlight, flooding into you in a torrent. People claim it's addictive.
And once that's over, once they come down off that high, some believe that a craving sets in. They describe a lust that sits inside, coloring everything they see, so that everyone - even the people they love - becomes a target, a potential source of that energy. They are addicted to a powerful drug but there are no limitations on where it can be found or how much they can have, because it's all around them, free for the taking, and their victims will be left only with a touch of fatigue, or a bit of a headache, if that - because the vamp is only taking a little, right? Just a taste? Some people do this for fun. They think that because they are capable of doing it, they have a right to do it.
Most people who believe in manipulatable life energy think that vampirism can be defended against, using techniques commonly referred to as "shielding". If you suspect that someone is feeding from you, it is generally suggested that you move as far away from them as physically possible. If you can't get away from them, then at the very least don't stand close to them and don't let them touch you (touch facilitates energy exchanges of all kinds).
Again, in the spirit of anthropological study, I've noticed that most people who hold these beliefs tend to be young - roughly teenaged through early twenties; those who are older tend to have these beliefs within the broader spectrum of neo-paganism. What used to be something rarely mentioned on the web over a decade ago has now become a common presence - there are even support groups. It's also strongly associated with otherkin. I suspect that the younger set represents a majority of the people who believe in psy-vamps because this gives them a way to feel unique and special before they have solidified their own personal identities.