Why is Lady Justice blind?

Because law is blind, should be blind to those to whom the law is being applied.

Our law should be impartial, and thus judge everyone equally.

We are entitled to equal justice under the law, according to our Constitution. We need judges who apply the law impartially and equally to all.

Empathy has no place in the judgement of law. Feeling sorry for someone, or feeling angy at someone, for their life circumstances or their beliefs or whatever, should not affect the ruling of law. If feelings do affect the law, then it is not "equal justice for all".

You wouldn't want the referee to judge one team more harshly than another team because the referee 'felt sorry' for the second team, or for a teacher to add points to a student's grade that the student did not deserve. And you don't want a judge handing down legal decisions based on any thing other than the facts of the case and the law of the land.