Spine exports and uses premultiplied alpha by default. It has some nice feature benefits and is pretty standard for games and video editing. That's why it uses it.
The Spine-Unity runtime comes with the Spine/Skeleton
shader which is a premultiplied alpha shader. You can use that so it matches Spine Texture Packer's default export settings and get rid of the black borders.
Unity's sprites are rendered with straight alpha by default. It's good for beginners to avoid errors and stuff, since it's a more common. It's also more immediately compatible with some Unity features like fog (I think).
Unity's Sprites/Default
shader (which Unity's Sprite system and Sprite Renderer component use by default) is a straight alpha shader.
To use the atlas exported by Spine, you need to export with the correct settings and render it with settings that match: premultiply on both, OR straight alpha on both.
See the examples here: viewtopic.php?f=8&t=3132