It seems, from a bit of googling, that people form Missouri have a bit of a reputation for being sceptical, doubtful and not easily persuaded: Wikipedia says:

"There is no official state nickname.[21] However, Missouri's unofficial nickname is the "Show Me State", which appears on its license plates. This phrase has several origins. One is popularly ascribed to a speech by Congressman Willard Vandiver in 1899, who declared that "I come from a state that raises corn and cotton, cockleburs and Democrats, and frothy eloquence neither convinces nor satisfies me. I'm from Missouri, and you have got to show me." This is in keeping with the saying "I'm from Missouri" which means "I'm skeptical of the matter and not easily convinced."

https://en.wikipedia.org/wiki/Missouri#Nicknames

There's a similar reference in the song "You came a long way from St Louis' too, reflecting the same "I'm not as easily impressed as most people" attitude:

"You came here from the middle-west
And naturally impressed
The population hereabouts
Listen, baby; I got news for you
I'm from Missouri too
So, naturally, I got my doubts"

So Gershwin's lyric seems to be saying "I didn't let cynical people put me off loving you"

John