It is clearly a predator, drawn to blood, but it also appears to not be a lot for it to prey on, particularly with large claws and teeth.
It must have been an early Christmas when the first breach opened and it found a target rich environment, but what did it do before that?
So far, of what we've seen in the Upside Down there are not very many good sources of food. This leaves open only a few possibilities for the Demogorgon, and we'll probably need to wait for Season 2 to see if we go back to the Upside Down and learn more.
The options that present themselves based on Season 1 are:
I don't know if the "demogorgon" itself existed before encountering 11, but it seems clear to me that the Upside Down did not.
Prior to encountering the creature, the place 11 was moving through seemed to be featureless black emptiness. It certainly wasn't the upside down we see by the end of the season, with mists and trees and such.
The final Upside Down seems to have been created recently, as it contains distorted versions of the current real world. If it has been around a long time, who built the dark Castle Byers?
People and the creature moving around in the Upside Down cause electrical disturbances in the real world, as observed when the mother and sheriff are walking around her house. If the creature and its prey were wandering around the area previously, you'd think someone may have noticed. Of course, maybe they did, and they are just ignored due to the tinfoil hat.
So, I think that when 11 touched the creature, it not only opened the rift, it pulled the creature into a new pocket dimension, which was formed based on the current state of Hawking, as well as the current state of wherever the creature came from.
If the creature had any prey where it existed before, assuming that it even did exist before, then that prey didn't come with it to the Upside Down.
Conclusion:
The Upside Down is a recently created bridge between wherever the creature came from and the town of Hawking. It displays characteristics of both places, and seems to continue to grow in detail and perhaps scope.
There may or may not be any prey there for the creature, but this is not a problem for the creature's prior survival because it hasn't always lived there.
From my observations its obvious that the monster was feeding off of an egg it had absconded with. It is not natural for creatures to devour their own eggshell, or even the egg shells of other animals for that matter. The sole exception being the toothless snake. There's also some other points made by others I thought I would comment on and they all feed back into the monster was eating something else's egg.
First, the mind-space that Eleven is seen in (pitch black emptiness) is absolutely and clearly not the sub-dimensional shadow realm (i.e. upside-down).
Second, the sub-space dimension clearly contains reflections of its parent dimension, and those reflections occur in real-time. It would be more accurate to say impressions (made by sine wave results of destructive interference) as opposed to reflections, based on what we can see of the static matter in the sub-space. I will touch on this below, but we don't see non-electromagnetically run dynamic matter being impressed on the sub-space.
Third, electromagnetic fields have real effects on others from sub-space. (Possibly this effect works both ways, though there is little indication of this thus far).
Fourth, powerful enough emotions or thoughts (possibly amplifying EM fields) are enough to cause a localized disturbance, or even dimensional merging/rifting (as evidenced by Will's most powerful contact with the mother causing a rift/merge behind the wallpaper in an early episode).
Fifth, it is with absolute certainty that Eleven herself is not "maintaining" an entire other sub-dimension. There exist reflective elements in the sub-space of its parent dimension of which Eleven has no knowledge, and are both detailed, extensive, and expansive in nature. While we may be able to agree with Eleven's conclusion that she helped cause or even solely-caused the initial large breach, even that may be something we can still rule out.
Likely this sub-space dimension has always existed, probably from the moment the parent dimension existed.
There is nothing to suggest that this monster is the only large-scale semi-intelligent creature that exists in this other space. In fact, things would suggest quite the opposite.
Once again to re-iterate: As far as the egg is concerned, there is nothing to suggest that the egg it was feeding off of was its own birth-egg. However, in the unlikely case that it was a "Demigorgon" egg, (being that fresh) would imply that it was recently born and devouring its own shell (which is not something that any animals are known to do). However, in nature we do find creatures that eat their other's eggs, or even self-produced eggs (e.g. Hens, fish, etc). That would be a far more likely scenario.
Regardless, this affirms there is other life in the upside down, whether in the form of an egg-laying Demigorgon, or something else. This is already confirmed anyhow, as we've seen other lifeforms (e.g. Slugs) in the sub-space which are clearly not indigenous to the parent dimension.
Interesting Notes:
The slugs have nothing to do with the Demigorgon's reproduction (in fact, in the original script they were going to use a Spider, but it wasn't "otherworldly" enough).
There is breathable air/oxygen in the sub-space. However, the particulate matter that permeates the air (in this area of the sub-space) is toxic. Apparently the toxicity is not fatal without extremely prolonged exposure. (Perhaps not even then)
There has been little sign of water, an arguably dynamic-matter substance. However, it could be argued that air is dynamic-matter in the same sense as well. (Or other gases and liquids). This suggests that dynamic-matter such as air or water is not impressed on the sub-space externally by the parent dimension, but rather produced internally. Just as our own Flora produce/recycle/convert our atmosphere, so too does the sub-space's own ecosystem.
Things to consider:
Considering the sub-space dimension has its own eco-system and history. (The creators have a 30+ page paper detailing how this shadow realm works, other areas, and its creatures) we may find that the demi-gorgon is the least interesting thing in it.
Where or what was the Demigorgon stealing the eggs from? Far too large for it to have laid them itself. If they are demigorgon eggs, then the adults much be terrifyingly huge.
Does Eleven attract other psychic-resonant creatures from the upside-down when she uses her telekinetic abilities? Or only her telepathic abilities? Even then, is it only when she enters her mind-space and projects herself out too greatly?
I assumed that it didn't exist until the first breach. It's created by 11 and in some ways is a part of her. It doesn't really need to eat to survive.
Personally, (spoiler warning for season 2) I think that the egg the demo gorgon was feeding on WAS a demo gorgon egg. I think the slugs escape from the eggs, possibly needing a huge egg because they may take years to develop and need a large yolk in the shell. They molt to have back legs, then front legs, then to be a demo dog and then, as far as we've seen, it stands on two legs and becomes a demo gorgon.
I think that maybe then, if the current mind flayer (which I consider a 'queen') dies, then the strongest demo gorgon goes into a cucoon or something and become the new mind flayer. This would explain how the egg could be laid IF it was laid for demo gorgon, it was laid by the mind flayer.
