Can finches survive in cold weather?

All birds will do this to keep warm.

When they puff their feathers out, air gets "trapped" within the feathers and warmed up by the birds body heat. It's like a home-made blanket.

If your bird is doing this regularly, make sure their cage is nowhere near a draft or a window which is constantly open, but also make sure they are not in direct sunlight, as extensive heat exposure will kill a bird.

You can also invest in a hanging bird bed. This is basically a tunnel made of fabric which is placed in their cage, the bird sits inside it to sleep while keeping warm at the same time.