As much as one can explain the reasons for an illogical supposition like this, here are the reasons why the WBC believes that the President is an Anti-Christ figure:

The Westboro Baptist Church feels very strongly about the issue of gay marriage.  They fervently believe that their interpretation of sacred scripture forbids the issue of gay marriage and that those who either participate in it or support it are essentially condoning sinful behavior. The WBC believes that the issue of gay marriage is the underlying reason behind so much that happens:

The Westboro Baptist Church blames gay marriage for the shooting of two firefighters in Webster, N.Y. on Christmas Eve. WBC spokesperson, Margie Phelps,  tweeted that the firefighters died for “fag marriage” and “other proud sin” on Dec. 27.

Prominent spokespeople from the church such as Phelps believe that with the President's statement in support of gay marriage, he has positioned himself as an anti-Christ figure.  Phelps has suggested that the President's position on this issue as well as his entire agenda moves him towards that position as identified in her interpretation of Scripture.  The group has articulated that the President's support of gay marriage is what makes him so much an anti-Christ figure. There is little other evidence that the group seems to suggest in terms of its position.  It makes the argument that the President is an anti-Christ figure because of his support of gay marriage, which they see as a type of "litmus test."  When searching for why the group believes the President to be such a figure, the group's website immediately directs to a page in which the statement against gay marriage is evident.  While it does not mention the President on this page, it makes the link clear between how it perceives the President through his position on the issue. Flawed as its logic is, this is where the group makes its stance on the issue.