What happened to the definite article?

I do not remember any specific comments by either Stephen Colbert or any of his production staff.

It is, however, not the first time that Stephen Colbert has renamed his show in reaction to ongoing events. During the 2007–2008 Writers Guild of America strike, Colbert, himself a writer, changed the pronunciation of the show's title The Colbert Report to include both "t"s, which are normally silent. He did this in solidarity with the writers and to emphasize the fact that without the support of his writers, the show is not the same show.

During the same time, Colbert's good friend and former boss Jon Stewart, also a member of the WGA and writer on his own show, changed the definite article of his show, The Daily Show with Jon Stewart to A Daily Show with Jon Stewart to show respect to the writers by making it clear that without their input the show is not The show but merely a show.

Colbert would have been well aware of this, not only because of his friendship and business relationship with Stewart (The Colbert Report was produced by his production company, also The Colbert Report was a spin-off of The Daily Show, where Colbert used to be a writer and correspondent), but also because Colbert, Stewart, and Conan O'Brien performed together on a series of ad-libbed skits crossing over all three shows during the time of the strike, with one evening even all three of them appearing on all three shows.

Even without an official statement, we can conjecture that the reason for changing the name is probably to express the sentiment that without the audience, without the Ed Sullivan Theater, and without the staff and crew, the simply is not the show that we are used to, but a different show.

During my research, I chanced upon a completely different potential reason. Apparently, Nielsen ratings depend on the precise spelling of the show name, and it is somewhat commonplace to intentionally misspell or change the name of a show in cases where lower-than-normal ratings are expected. This tactic is, for example, employed on shows that air on major holidays or opposite major sporting events. That way, the show gets counted as a completely different program and the ratings don't impact the ratings statistics for the show.

There is no reason to believe that this specifically is Colbert's motivation for changing the show, but it might be something that the network would have considered changing the name anyway. Although even if that were the reason for changing the name at all, the question of why this specific name change would still be just as valid, since any misspelling would have sufficed. (The linked article mentions NBC misspelling the "Nitely news" on Memorial Day weekend as an example.)

They could have called it The Late Shøw, The Laid Show, The Latte Show, or any of a myriad other variations, but they specifically chose A.