It's less that money magically changes minds, and more a few other more important factors:
1) Awareness raising. A lot of people are partially or completely unaware of what laws are on the ballot, what initiatives actually mean and what the stakes are.
2) Driving turnout. This is actually probably much more important than #1. If you look at the statistics, the percentages of eligible voters who actually vote in elections is not incredibly high. Money can be used to ensure that people who are sympathetic to your cause are properly registered to vote, know where their polling places are, even literally give them transportation if they don't have it. For this issue, to take an example, it is extremely likely that college students are overwhelmingly for gay marriage, but a lot of them are not yet registered to vote, or are from out of state and haven't re-registered in Washington, so this kind of money lets pro-Marriage organizations have a heavy presence on campuses getting kids registered in the state, telling them where they need to go and when to vote, possibly even renting busses or vans if polling places are inconvenient for campuses.
With marketing. [Obviously subjective opinion ahead that reflects my personal take].. In California, Prop 8 (Same-Sex Marriage Ban) won because of money spent convincing the middle (people who frankly didn't really care much one way or another) that if gay marriage was legal then we'd start teaching kids to be gay in schools, or something. I'm not kidding.
But yeah. 'Round here laws are products, and marketing is a huge part of how they're sold.
Legislators aren't omnipotent. They don't have a comprehensive view of what the electorate is thinking. If Amazon hires some lobbyists to do surveys and polls, and comes to a state legislator saying "65% of residents and 85% of businesses in your district support this bill" that's enormously persuasive to the legislator.
