A more thorough explanation:

Definition: Provided is a conjunction that means "on the condition or understanding that" or "except that".

1. We will sign the contract provided that you agree to the following conditions.
2. All permittees must be at least 18 years of age, provided that those with a bona fide hardship must be at least 15 years of age.
3. A railway car must be operated by a full crew if it extends for more than 15 continuous miles, provided that a full crew must consist of at least six railway workers.

The first example shows that the contract will only be signed if the other party agrees to the conditions. The second example shows that there is an exception to the age requirement for permittees who have a hardship. The third example shows that a full crew is required for a railway car that extends for more than 15 miles, but there is an exception that the crew must consist of at least six workers.