-To allow one more latitude or freedom than usual; to be lenient with one.
-Give additional freedom to someone
-Not judge someone severely
-Give someone a break
-Make some allowances for someone
-Allow a person to something that is usually not allowed
The origin of this phrase can be alluded to the docking of ships, where give me some slack meant to loosen the rope.
A similar phrase, with a similar meaning but slightly different form – cut slack for- was used in 1855 by Frederick Douglas in his book My Bondage and My Freedom.
Example
Oh, you know I never make requests like this, cut me some slack
I was really busy last week and couldn’t finish the report. Cut me some slack and I’ll do it by tomorrow.