According to the traditional understanding, the Satan shows up on Rosh Hashana with a list of all our misdeeds wherewith to prosecute us. He bases himself on the daily shofar blowings. When we suddenly stop, he gets confused that perhaps he missed the date of Rosh Hashana. Therefore, he does not show up the next day, which in truth is Rosh Hashana, and cannot, therefore, accuse us of anything.
It's a lovely answer, but following its logic, we are left with more questions. Most importantly, why doesn't the Satan figure this trick out already? Year after year, he falls for it. Strange, no? And why does he bother calculating by the shofar? Why not just Google the date of Rosh Hashana?
There are some other answers out there that are clearer. One is that the Satan bases himself on our reactions and fears. If we are blowing the shofar, we must be nervous. We must be going that extra mile because we've really been bad. That encourages him to accuse us even more. When we suddenly stop on the eve of Rosh Hashana, we exude confidence that we've already repented and atoned, and therefore have nothing to fear. That confidence is what confounds the Satan.
I love this answer, because it is so psychologically true and important. What is notable about Rosh Hashana is the absence of "confessional" prayers and petitions for forgiveness for a list of sins. Instead, we pray for global things, such as peace, prosperity, awareness of God and the advent of the Messiah. We remove the focus from sin and put it on to perfecting the world.
The Satan cannot prosecute us if we are allied with God in dreaming of and working towards a perfect world. We've taken the discussion to a level that the Satan cannot reach. It is similar to dealing with a parking ticket when you're a presidential candidate and you have to go to a nationally televised debate. Compared to that, parking tickets are small and unimportant. We basically "out-league" the Satan starting on the eve of Rosh Hashana.
I believe that this explanation can also explain the "confused date" explanation. The Satan deals in parking tickets. He expects that Rosh Hashana will be the mother of all parking ticket hearings, and builds his files accordingly. (He confuses it with Yom Kippur, when we deal with sin. I could even suggest that the Satan expects that Yom Kippur should precede Rosh Hashanam at least thematically.) When we stop the Shofar on Rosh Hashana eve, we are telling him that we don't deal with parking tickets any more. We're running for World-Fixer, and need to prepare for THAT debate, which the Satan can't even get in to.
That's what it means by him being confused, thinking that he missed Rosh Hashana. He's thinking that, if they stopped blowing the shofar, their sins must have been judged and acquitted already. Beyond that, he has no role.
In truth, though, we DO need to deal with the parking tickets. We do it especially on Yom Kippur. Nonetheless, before we do that we must focus on what's IMPORTANT in life and the world. We focus on the WHY of Judaism, the destiny of our nation, before we deal with the HOW, the scorecard of our personal behavior.
So let us put the clear goal of fixing the world and ourselves before us this Rosh Hashana 5770. Then, towards Yom Kippur, we'll bring this into our personal lives by improving our character and behavior accordingly.
A shana tova, a year of Life, Health, Happiness and Prosperity to you and yours!