What makes the Hobbit Holes look so cosy?
In addition to the elements mentioned in User18849's excellent answer, I think the simplicity of each element (in terms of usability and function) and obvious quality of everything, greatly contribute to the apparent coziness.

This hobbit hole is an unintimidating space where you don't have to worry about figuring out how to use anything, nor that you might accidentally break an element because it is flimsy or cheap. This is a space that can be lived in without effort or disproportionate carefulness. 

Additionally, the walls, doors and even the windows seem thick and solid; which reinforces a sense of safety from the dangers outside