This claim doesn't appear to have come about as some sort of misinterpretation of a constitutional process or other legislative pathway, but as a predicted response to the publication of information he claims to possess which will supposedly make President Biden & Vice President Harris' positions untenable.
Lindell first made his August 13th prediction on July 4th - at the same time he confirmed the dates he plans to host a "cyber symposium". During the event, held between August 10th and August 12th, he claims that he will reveal "the cyber data and the packet captures" - supposed proof that President Trump was the true winner of the November 2020 election. There appears to be no other attempted justification for the proposed date of August 13th except for it being the day after the conclusion of this event.
Of course, even if Biden and Harris were to resign, the presidential line of succession would mean that the Speaker of the House of Representatives would become president, not Donald Trump.
Speaking on Real America's Voice, Lindell said:
Once we have the symposium, by the night of the 12th or the morning of
the 13th, if everyone has seen it including the administration that's
in there now that didn't win, maybe, you know, Biden and Harris would
say 'Hey, we're here to protect the country' and resign. And, you
know, I'm serious!
The Sun also reported on his claims:
Lindell's speculation comes as he vows to showcase proof the evidence
was stolen during his "cyber symposium" event, which is conveniently
scheduled for August 10 through 12.
"It gives me hope that even your Democrats," he said. "They're seeing
now firsthand what government can do and how bad socialism and
communism can be."
"When we get through this and the Supreme Court pulls down this
election -- like I've been telling everybody -- when they do this,
it's going to be a great uniting and that gives me hope."