The expression “old familiar stamping ground” means: A place to which one is accustomed.
Back in the Revolutionary period, and probably long earlier, a stamping ground was a place known to our American forebears where horses or other animals gathered in numbers.
The step was short in a transference of the term to a place to which a man, woman, or child was accustomed.
First to use it in a published work was H. R. Howard, compiler of The History of Virgil A. Stewart (1836) : “I made my way from Milledgeville to Williamson County, the old stamping ground.”