To make a long story short, it appears as though two things happened:
- A man convicted of the death penalty gave a moving speech (or perhaps sang a hymn) which convinced the crowd that capital punishment was a travesty.
- At roughly the same time, Canada put a Detroit resident to death for a crime which was later confessed to by another man on his death bed.
I guess the biggest takeaway here is that Michigan was very sparsely populated into the 1830s. Although it was rapidly settled after statehood (Wikipedia notes the population booming from 80,000 in 1837 to 212,000 in 1840), you would still have had a state government strongly influenced by the opinions of the few who had been there before incorporation. The above events took place in 1830 and before 1835 (the article cited above notes the deathbed confession having occurred in 1835), so it appears that when Michigan did become a state, there was a strong thread of "we are an anti-capital punishment state" already there.
Incidentally, Michigan was also not alone in abolishing the death penalty during this time frame. Wisconsin and Rhode Island also passed statutes which were, in a sense, even more stringent than Michigan's inasmuch as you could still be hanged there for treason. Elsewhere, moves were made at about this time to end mandatory death penalty sentences for many crimes (across the pond in England, the Bloody Code saw death penalty sentences for crimes such as grand larceny and pickpocketing, the latter of which created a queer situation in which a crowd that showed up at a pickpocket's hanging could themselves be visited by a pickpocket working his trade).