Why is the white group marked as black?

Why do the white stones and territory in the bottom right count as score for black?

This group has only one eye. A group that is alive has always two or more eyes.

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No you can live with 2 fake eyes

(Or even with seki and no eye, 1eye…)

Black can capture white by playing in the 2 spaces. If white capture the black stone, white can play again and capture all.
If white don’t capture, black still play again and capture all.
Try it.

As @Atorrante said, I suggest you to go check what are eyes any beginner tutorials should do the trick.