Question: How will the world (especially the Antichrist) explain the Rapture—particularly in light of so much publicity around the Left Behind series?
Response: The sudden mass disappearance worldwide of 100 million (or maybe more) of earth's inhabitants will cause a panic beyond anything we can imagine. I believe that all of the babies and small children (of Muslims, Hindus, Buddhists, atheists, etc.) will also vanish, taken to heaven at the same time.
No matter how familiar with the Left Behind series, whoever "received not the love of the truth...believed not the truth...God shall send them strong delusion, that they should believe a lie: that they all might be damned..." (2 Thes:2:10-12). I don't think that necessarily means they have specifically rejected the gospel of Christ. It could also mean they have rejected the witness of creation and conscience.
Neither the Left Behind series nor any other books or DVDs will benefit any such person. Of course, hundreds of millions (almost all Muslims, most Roman Catholics, multitudes of followers of other false religions, and atheists) who never knew and rejected the truth will have the opportunity to believe and will be martyred for their faith under Antichrist.
Antichrist's explanation of the Rapture is not important. What is important is the fact that all will be deceived by it who oppose the truth.