Pets change the biochemistry of their owners. Thid is a fact. Petting an animal may temporarily increase some hormone levels relates to joy and love.
I'm serious. Here is some SCIENCE! to back it up:
Among the well-documented effects of HAI [Human-Animal Interactions] in humans of different ages, with and without special medical, or mental health conditions are benefits for: social attention, social behavior, interpersonal interactions, and mood; stress-related parameters such as cortisol, heart rate, and blood pressure; self-reported fear and anxiety; and mental and physical health, especially cardiovascular diseases. Limited evidence exists for positive effects of HAI on: reduction of stress-related parameters such as epinephrine and norepinephrine; improvement of immune system functioning and pain management; increased trustworthiness of and trust toward other persons; reduced aggression; enhanced empathy and improved learning. We propose that the activation of the oxytocin system plays a key role in the majority of these reported psychological and psychophysiological effects of HAI. Oxytocin and HAI effects largely overlap (...)
Cats are a special case. I feel confortable living with one because before she came into the house, any weird noises at 3AM could be the demons I often end up summoning from the dark dimensions whenever I am try to sing songs in foreign languages that I don't speak. Now I can go back to sleep feeling safe because I know it's just the cat knocking stuff off shelves.
Seriously though: cats harbor a parasite that only reproduces sexually inside them, but which they may pass on to any other mammal. Every cat owner has it in them. Some scientists believe these parasites can alter people's bebaviour, but there is little evidence. Do check the wikipedia entry for crazy cat-lady syndrome, though. I remember some papers on how infection by this parasite makes some mammals more tolerant to the smell of cat urine. I found one for rats, and this paper says (emphasis mine):
Latent toxoplasmosis, a lifelong infection with the protozoan Toxoplasma gondii, has cumulative effects on the behaviour of hosts, including humans. The most impressive effect of toxoplasmosis is the "fatal attraction phenomenon," the conversion of innate fear of cat odour into attraction to cat odour in infected rodents. While most behavioural effects of toxoplasmosis were confirmed also in humans, (...)
So what is my long rambling about cats and their owners' behaviour about? It's just to point some hard evidence that frequent contact with specific species cause specific changes to humans' brains and/or hormone levels.
Suppose some species of tapir harbors a parasite which only reproduces inside them, but which may latch onto humans. This parasite may interfere with Rapid Eye Movement cycles during sleep, which is when dreams become more vivid and easier to remember. Or they may cause dreams to not happen at all due to some unknown mechanism.
People who have recurrent nightmares will want this tapir for a pet due to this effect. They might also become as common in hospital wings as dogs are becoming nowadays, though the tapir would be more noticeably associated with psychiatric patients.
The thing the tapir gets from usage of its "power" is the same thing every domesticated species got: shelter and food. The dream-eating tapir is now so dependent on humans for those that of you release them into the wild, they die. Either they can't find food, or they become it to some predator.
About the satisfy-the-creature-or-lose-your-hopes-and-dreams part: a healthy tapir will have healthy parasites. If the tapir is undernourished though the parasites will activate some genes which cause their effect on humans to be nasty. The human will have increased levels of cortisol and adrenaline, causing stress and restlessness.
As is usually the case, humanity would have started domestication of the dream-eating tapir well before science became a thing - so any attempt to explain the tapir's power will get you into the territory of myths and legends.