This phrase often implies a level of trust in the person given freedom. The librarian gave me the run of the place, knowing I'd handle the books with care.
Pro Tip 2/3
Not Literal
It doesn't necessarily mean you can do anything; there are still unspoken rules. I was given the run of the house but understood not to enter the study.
Pro Tip 3/3
Responsibility
With the freedom comes the expectation to respect and care for the place or situation. Having the run of the campsite, we ensured to leave it clean.