why is the full faith credit clause-regarding same-sex marriage a very influential federal issue and why?

Answer 1

It has been used to fight against DOMA ( though not successfully).

Answer 2

Wrong tense. It was a big federal issue back in the 1990s when the Defense of Marriage Act was passed (around the time that Hawaii momentarily recognized same-sex marriage). DOMA has never been successfully challenged and the general legal consensus is that DOMA is within Congress's authority under the full faith and credit clause.

Of course, since same-sex marriage became an issue in 2004 with the decision in Massachussetts, the possibility that DOMA might one day be successfully challenged has become a boogeyman used by those who want a federal constitutional amendment on this issue. It is not a substantial issue but it is used as a way to pander to and scare a significant number of voters who oppose same sex marriage. I would not, however, say that this proposed constitutional amendment has become an influential issue yet. That, of course, can change since a week is a long-time in politics, but it doesn't appear to have become a separate voting issue (aside from the general panorama of family values issues).