Most Likely Due To Jurisdiction Issues, Prior Case Law, First Amendment Rights And How Contract Law Works
To me, this seems quite straightforward: Musk's original (now-deleted) tweet reportedly offered million-dollar checks on the condition of voting.
His statement on Twitter wouldn't meaningfully constitute a legally binding contract, because it doesn't offer anywhere to 'accept' or 'sign' in any meaningful capacity. For example, if a man bombastically says to a group of friends 'A million bucks says you can't do it', it does not automatically constitute a legally binding contract; at best it is an 'offer'. An offer can be withdrawn at any time, so long as he hasn't accepted anybody wanting to accept that particular offer.
What is the legal basis for this challenge being rejected by the three courts?
We can learn more insights by examining the case in Pennslyvania where judge Angelo Foglietta explains his reasoning (in this case it relates to if it counts as a lottery but we learn a crucial detail from it):
Musk had repeatedly explained online and at rallies that those chosen to receive $1 million during the 18-day giveaway were being asked in return to be spokespeople for America PAC, the judge wrote.
The PAC's treasurer "testified credibly when he stated that those selected to earn the one million dollars were selected by the organization in a multi-step process that involved looking at their public posts on social media and meeting them in person before the town hall events to make sure their personality would be a good fit for the role," the judge wrote.
https://www.businessinsider.com/elon-musks-1m-swing-state-sweepstakes-was-never-a-lottery-2024-11?op=1
The actual legal contract, as the judge saw it, contractually offered something else that isn't voting, testified on legal record: the person works for Elon Musk's PAC, and it involved speaking at 'town hall events'. Statute 12.11(3)(a) explicitly states it does not prohibit someone from stating their personal political preferences, so spokespeople would have been fine.
Also consider free speech rights re: First Amendment. Elon Musk does not reside in Wisconsin; a thorny issue of whether or not Wisconsin could prohibit such speech for a man not resident in the State. Payments weren't made by Elon Musk, but by a PAC, meaning he might not have even entered the State to become liable in terms of jurisdiction at the time he wrote the post. It would have a chilling effect if one State's laws could impact another State's (or all States') laws and citizens, and may violate Interstate Commerce if it interferes with how some businesses (or PACs) work.
Section 441b's prohibition on corporate independent expenditures is thus a ban on speech. Justice Anthony Kennedy, Citizens United v FEC
We have case precedent Citizens United v. FEC decided by SCOTUS which found restrictions on political spending for corporations and unions by the FEC were deemed to be violations of the First Amendment. Whilst this 'only' covers corporations and the FEC, it expanded rights to include corporate entities (read: PACs). Whilst we don't know for absolutely sure, if Musk had appealed to SCOTUS (he can afford high powered lawyers), odds are overturning parts of Wisconsin law would have been a very likely outcome.
https://supreme.justia.com/cases/federal/us/558/310/
Elon Musk likely didn't want a legal battle (expensive/time consuming), and Wisconsin likely didn't want to set the precedent that their laws might be in violation of the Constitution (which would explain why the Supreme Court declined to comment). So rather than fighting a man rich enough to defend himself in court, they probably erred on the side of caution and let it pass.
Unfortunately this answer is speculative, but with the judges not disclosing their reasoning, given the evidence available, it seems like a reasonable interpretation.