This is difficult but possible. Birds are easy because they have smooth outlines, while fur makes things more troublesome. There are two routes I would take when editing the image: raster or vector. I personally use Adobe Photoshop and Adobe Illustrator respectively for these, so I’m not sure whether other programs have similar features.
By using the Magic Wand tool, you can select chunks of the background and delete them one by one. The outline details aren’t perfect, but it more or less does the job. Here’s the end result:
Adobe Illustrator has this thing called Image Trace that can turn raster images into vector. Turn the settings to Grayscale, with number of Grays at the maximum, 100. Then Expand the image, then Ungroup. The result is like a mosaic made up of thousands of tiny blobs, each their own solid shade of gray. After that, just select the chunks of background and delete them one by one. However, it may take longer due to the small size of each individual blob.
I personally prefer the raster method, as it’s easier, faster, and most raster programs can do it so you don’t need to buy Photoshop.