How is the age of Giant Tortoises determined, verified, since they normally outlive their human observers?

They generally keep records and identify the individual somehow, for example by marking it (such as filing notches on the edge of the shell, or putting an ID  tag on it. If the tortoise lives in a zoo, then the zoo knows when it was acquired, how big it was when it was acquired and how long has it lived in the zoo. 

Many have been kept in captivity for multiple human generations..  The records would be there from the people or organization that previously housed and cared for the animal.

You should talk to your vetnerenerian if you want to know the age of your giant totoise,

I had a giant tortoise, and he lived to be 113 years old! My grandmother gave it to me, and the veternarian lived him :)