There are two ways to answer this question.
First, we can ask why money is allowed to be such an important factor in American elections. The reason for this is that the Supreme Court has ruled that spending money on campaigns is a form of free speech. For example, the recent ruling in the Citizens United case opened the way for corporations and unions (as well as individuals) to spend as much money as they want on political campaigns so long as they do not officially coordinate their efforts with any individual candidate’s campaign.
Second, we can ask what value money has in elections. Money is valuable and important in elections because so many aspects of a campaign depend on money. The most obvious way in which money is useful is for buying campaign ads. We all know that candidates air huge numbers of advertisements and money is necessary to make the ads and to buy airtime for them. However, this is not the only way in which money is important. President Obama’s recent reelection has been credited to his “ground game” which engaged in get out the vote activities. These cost money too as the campaign must pay for computer hardware and software for identifying likely voters and for hundreds of field offices from which to try to win their votes. These and many other activities cost a great deal of money and can be extremely important to the result of an election.