The "Galileo Affair" was obviously a pretty funny situation in general. The Catholics were losing so much territory in Europe at that time to Protestantism (relax, it is only England) that they were extremely paranoid about even seemingly insignificant "heresies". Although it is a myth that Galileo was tortured for his views, or even excommunicated, he was condemned and imprisoned for teaching heliocentrism (although it was more like house arrest).
It's quite unclear why exactly it is heretical, of course, to say that the Earth orbits the Sun. The bible sort of suggests that the Sun goes around the Earth, but it isn't some kind of central point about the Earth being the center of the universe because God created Man as the center of all things, like some people seem to think. It's more like, the Bible mentions one time how the Sun stopped in the sky, and that sort of sounds like it means the Sun is moving around the Earth. Not only that, but the Bible probably only takes that view because the people who translated the Bible had that view from Ptolemy, so it isn't so much the Bible being overturned, but Ptolemy. When Copernicus himself presented his ideas to the Pope, a hundred or so years earlier, he was awarded with a gift! But like I said, the Church was very insecure at this time, so I guess they couldn't mess around.
This lead to some pretty funny situations, like some Catholic philosophers refusing to even look through Galileo's telescope, presumably because they were afraid they might themselves get convinced by the evidence, and then be holding heretical opinions. Not much of a philosopher, apparently.
The Pope had actually given Galileo direct permission to argue for heliocentrism, as long as he didn't take a position. Galileo, unfortunately, made the argument far too well, and it was obvious that he believed in heliocentrism. Like the comic suggests, it seems to me that anyone making this argument must have been convinced by Galileo, and therefore heretical themselves. Unfortunately Galileo didn’t take up this argument, and merely denied taking a side on the debate, and also that it wasn't heretical because the bible itself doesn't take sides on physics (two contradicting arguments at once Galileo, not great rhetoric!)
Probably my favorite fact though is that one of his punishments was having to recite seven psalms of penitence once per week, but his daughter Maria Celeste petitioned the Pope to have it count if she did the prayers for him. Presumably this freed up more of his time to...I don’t know, think up more heretical opinions, presumably?