It seems obvious when you put it in as little context as we are going to here, but everyone likes blankets. How could you not? They provide life sustaining warmth and a degree of comfort that the open air could just never afford. Why your dog likes blankets can get a little more abstract then that. Though there are countless videos online of dogs taking a blanket and laying it over them, and to an adorable effect, it is far more common for dogs to simply lay on them. Regardless of how they interact with the blanket though, they certainly seem to gravitate towards them. As blankets would hold no part in their natural habitat, obviously, it seems unusual for this to be such a common behavior, let’s dissect it.
The Root of the Behavior
Does your dog like their crate? Do they rush to it when the vacuum turns on and they run in terror? It turns out these behaviors are pretty closely related. In the wilderness, especially in hostile environments, dogs create a den. These dens provide them shelter from harsh storms and frigid temperatures. They provide protection from predators who would see them as prey and a place to call home where their family could congregate. In this sense, it is kind of hard-coded into their genetics. A blanket is not the same, of course, but the same impulse that drives them to their kennel or pet bed when they are frightened is the same impulse that enjoys the warmth and soft textures a blanket provides. They are seeking comfort, and a comfortable environment in this context is a place that is safe, warm, and familiar.
Stress and anxiety can exacerbate and amplify these behaviors. A stressed or scared dog is likely to immediately retreat to the environment they are familiar with that they have deemed safe. For this reason, it is important that your dog has some area in your home that is for them. It does not need to be a whole room, but simply a dog bed or a pile of pillows and blankets that they can call their own. Many people use their kennels for this purpose, and many dogs prefer it that way too. Having this place to retreat to when frightened will provide a great amount of relief for your dog. In addition, this small space becomes a home for them where they know they can go rest or relax relatively undisturbed. To ensure this place is always a safe a comfortable place for them, try and keep it out of the major walkways in your home and away from where people and excitement congregate.
Encouraging the Behavior
Some of this can be a learned behavior. Like human babies, for pups, the feeling of being tightly wrapped in a warm and soft blanket provides a great sense of security. Like a set of armor around you, perfectly molded to your form. With knowledge of that, people often swaddle pups in blankets like they would a newborn child. It could be that a few such instances in their early years created a correlation between a blanket and a safe environment. This correlation can follow them through their life as they reinforce that behavior themselves by returning to that place whenever they seek comfort. If you want them to stop burrowing into your blankets then you simply need to provide them another place they can get that comfort. You could likely train them not to do it regardless of you providing them another area, and a professional trainer using Pavlovian techniques could certainly accomplish this with relative ease, but for your companions sake, provide them another place. A trainer could equally as quickly turn his kennel into that place, and your dog wouldn't fear their kennel, but love it and go their often of their own volition. A zone like this can prevent anxiety issues from forming and keep your dog in a healthier mental state.
Other Solutions and Considerations
This behavior is more common in some breeds then others, and this would exclusively speak to their heritage. Smaller dogs, ones that would make easier and more common prey if they were in the wild, exhibit this behavior at a higher percentage then larger breeds with only a few exceptions. For example, dogs like Malamutes and Huskies, which are accustomed to the cold and harsh climates found far to the north. These breeds, and breeds from similarly inhospitable climates, do this frequently. For them it was out of a need to be protected, not from predators, but from the elements. For all the comfort it provides them, it provides humanity with far more. Nothing is as cute as a husky pup diving under the blankets.
Unless these burrowing behaviors move out in to your yard, it is probably just adorable. If it has moved out into the lawn, get them a spot they feel is comfortable in your home that they can retreat to. If that continues, contact a trainer or an animal behavior specialist to learn more about what is causing this and what you can do!