Woman: Discrimination against women is wrong!
Man: But ma'am, don't you think that you're a little biased?
I have been accused of being "biased" in some of the causes I advocate (even by my most supportive of fans, if in a loving sort of way). And my opinion has always been that a) everyone has biases, b) people are generally drawn to the issues that their life experience biases them towards, and c) there is nothing wrong with that.
Now, it's obvious that a person's biases can blind them to reality - and I think this is the fear at play when people accuse others of being biased (unless they're just trying to discredit someone because their views oppose their own biases, which is not uncommon).
And, as a result, I think it's important to - not so much examine the person's biases, but to examine the argument they're actually making and its legitimacy. Focusing instead on the bias, rather than the logical effect that bias may have on the argument, is a form of the ad hominem logical fallacy - attacking the person instead of their argument.
If the argument is invalid (perhaps inspired by the person's own biases), then that should be clear in the argument itself. If, on the other hand, the argument is a good one, then it shouldn't matter what the person's biases are. And, indeed, bias can be just as much a tool for righteous social change as it can for the propagation of bigotry.