That full faith and credit shall be given does not, in itself, mean that the states are required to alter their laws to accomodate the documentation from other states. It only means that, consistently with their own laws, the states must treat documentation from other states the same way it treats its own documentation. Hence, a marriage license from Hawaii between two men is only good in California to the extent that its laws allow men to marry each other.
If certain restrictions on marriage (such as age restrictions) can be evaded by going to Nevada or Utah or someplace like that, that's only because either the other states have agreed to allow it, or because Congress has mandated it, pursuant to the second sentence of the Section that you quoted.
I'm pinging our resident legal expert, just in case I'm mistaken about this, in which case I'll just have to hit the books again.
The clause is self-executing, no Congressional legislation is required.
Congress has legislated under the clause, however, in the Defense of Marriage Act, which is designed to block the scenario I outlined. I just don't see how the DOMA can survive the reasoning of the recent case. On the other hand, O'Connor doesn't see the need to employ consistent reasoning from case to case, so DOMA might be safe after all.