The text of the amendment states that its purpose is:
To recognize, for purposes of the Immigration and Nationality Act, any marriage entered into in full compliance with the laws of the State or foreign country within which such marriage was performed.It does so by saying that any marriage (including same-sex marriages) which are legal n the jurisdiction in which they were performed will be recognized as a marriage for the purposes of immigration law. Importantly, the marriage would still be recognized for immigration purposes even if the state in which the couple lives or moves to does not recognize that marriage as legal.
Lavi Soloway of the DOMA Project explains implications of the Leahy amendment:
With this bold move, Senator Leahy has carved out an exception to the Defense of Marriage Act for lesbian and gay binational couples that will provide access to existing marriage-related family unification provisions of our immigration law. The Leahy amendment does not actually amend any current provision of our immigration law, but simply removes the extrinsic barrier caused by DOMA that prevents lesbian and gay Americans from filing petitions for their spouses, fiance(e)s and stepchildren. The implication of Senator Leahy's focus on equality is that LGBT families are no different than any other American families comprised of citizens and non-citizens.The unfortunate part of not having the amendment pass in committee as opposed to on the floor of the Senate is that in committee it needed a majority vote (which could have been provided by all Democrats) while on the floor it will almost certainly require 60 votes to be included in the comprehensive immigration bill working its way though the Senate.