It's like saying "This is..." expect the word order is reversed here. You can say:
Taniec to moje hobby = Moje hobby to taniec. (to + nominative),
Taniec jest moim hobby = Moim hobby jest taniec. (jest + instrumental).
You should be already familiar with these two forms of the sentences on the left (This is... construction in Polish). I can explain it briefly if you are not.
I suppose it makes sense but I don’t get why reverse it like that. Also, doesn’t hobby decline?
The inversion is usually used to stress the word, in this case hobby. Here, however, it rather changes the meaning somehow. You can compare it with Dancing is my hobby vs My hobby is dancing. I think the difference in meaning is almost the same in both languages.
No, it's the same in every case, both for singular and plural (Moje hobby to taniec i muzyka).
Bo "taniec jest moim hobby" :D Moim hobby is in the instrumental case, but the order was switched, that's all :)