I was probably a child when I last read it, so I might have some details wrong, but here's how I remember it:

A child is given a toy rabbit. A fairy visits the toy rabbit and gives it the gift of awareness. The child and the toy bond with each other and grow to love each other. Unfortunately, the child becomes dangerously ill, and after the sickness their possessions must be incinerated to prevent contamination. This includes the toy rabbit. However, the fairy arrives at the last minute, declaring that because the rabbit learned to love it was therefore a real rabbit, and with a wave of her wand transforms the toy into a living being and whisks it off to the woods were it lives happily ever after with the other rabbits.

So I guess my question is this - Do you think the velveteen rabbit and the fairy are real? Or is the fairy's magic an invention of the child's mind?

I think the narrative required the velveteen rabbit to be burned because it was so horrible. To the grown ups it's just velveteen, but to the child it's a dear friend. Even as children we know that being burned is horrible. So the child invents a solution where their toy can live happily ever after even after it's thrown in the fire.

I think there's definitely some Heaven and Hell symbolism to be had too. The velveteen rabbit was damned to hellfire unless it accepted love into its heart during its life. Then it is granted into the afterlife. In fact, you could say it was reincarnated into a higher spiritual form.

The story explores coping with loss as seen from the point of view of a child. Even though the velveteen rabbit was just a toy, the child has given it a soul. If you have a soul, when you die you go to the afterlife and live happily ever after. It's a comforting story to a child, and one that many people around the world have believed throughout the ages.