In American politics today, political parties have all the power. It is almost impossible to get anywhere on a larger level without belonging to one of the two parties. There is another reason to join a party though. Starting at the grass roots level as a volunteer, you can learn a lot about the ins and outs of politics.
All of this is true, and I need to add one other important element--money. Many candidates, if not all of them in the "official" parties, do receive monetary benefits in the form of advertising and other campaign elements. Unfortunately, we live in a political environment in which it takes money to make money and garner votes. It takes other things too, of course, but the foundation is always money.
The main benefit that a politician gains from joining a political party is a built-in grassroots support system that extends all the way through the national offices. Rather than having to start from scratch and build a constituency and seek resources, the politician can access the existing system.
However, being identified with one of the political parties can also be a politician's biggest handicap at the same time. Being identified with a particular party can polarize voters against a candidate, regardless of how worthy his or her platform may be.
For better or for worse, our two party system is pretty locked down. We have occasional third party "reform" movements, but they usually fragment and bicker away to nothing in one or two election cycles. The parties are big, and they can fundraise and everyone knows them, so they tend to preserve themselves. By being a part of either party, you can get yourself access to that organization/machine, and the money you would need to get elected. It's not a perfect system, that's for sure, but it's the best one we have.
In the United States, it is difficult to break in to politics if you do not join a political party at the local level. If you have enough money or you are already famous, it is possible, but it helps to join a party.
The only reason that joining a party is really all that important is networking. If you join and become more important in the party, you will meet up with people who are influential and who may have various kinds of expertise that can help you. Parties can help you meet more people and get them to support you for various political offices. These kinds of people might then go out and campaign for you by talking to their friends. Parties can also help you with campaign strategies, getting in touch with potential donors and things like that.
However, you can also get into politics in other ways. This is especially true at the local level where many (if not all) races are non-partisan. In such cases, you can get into politics simply by being well-known to influential people in your community.