Why did Nelson Mandela fight for democracy?

Because he belonged to the group that was strongly disadvantaged by this system.

Apartheid set whites and blacks apart in society, strongly benefitting the whites (in giving them all the real and legal power) and strongly obstructing life and blocking chances for wealth, power and success for blacks.

And Nelson happened to be born black in South Africa. So there you have a strong personal motive to fight Apartheid.

On top of that , Apartheid did not meet his own moral criteria of how a humane society should be (as it didn't meet the moral criteria of thousands, millions of other people around the globe anymore in the 70ies, 80ies seen the worldwide protests and embargos).