Visibility via advertising puts the candidate's name in people's minds. That way when people go to vote, assuming they're not the typical mindless drone that just goes down the ballot and checks all the democrat/republican boxes, they vote for names that sound familiar. Because they're more likely to pick a name they've heard of than one they haven't. That's how you buy votes.
I don't buy it.
First, because straight-line voting is so common.
Second, because, empirically, campaign funding (when both sides have any meaningful sum) has little to no correlation with victory in swing-states or areas with roughly equal party registration.
Third, voter turnout is very low in the US, especially among uneducated people. The average American is very politically uneducated. But the average American voter (not potential voter, but actual voter) is significantly better educated than the average American. Some, even many, may be greatly swayed by simple recognition, especially in low-level elections for county and state offices. But I think most, especially for national offices, make decisions based on bounded understandings of policy, largely gotten by word of mouth and ads. Racial allegiance, religious influence, family history... are all much bigger influences than campaign funding.
Fourth, campaign funding (or threats of funding the foe) DOES have an influence on post-election legislative choices by congressmen: which seems most easily limited, not by complex and easily circumventable campaign finance rules, but by longer, staggered terms and reducing incentives for rent-seeking.
