Maybe you need an illustration for the game that you made, or for the cover of your novel; or you need editing for your book, graphic design for your new ad campaign, or development work on your website. Whatever your specific project is, the process of hiring freelancers to help you finish it are largely the same.
There are two main ways of doing this.
Search out freelancers you like and contact them directly. Every artist, writer, and web developer in the world has a portfolio website, and you can look through all of them to find what you want. The benefit of this method is that you can get exactly the skill and style you're looking for. Unfortunately, there is no guarantee that the person you're contacting will respond to you, or be within your price range. If you don't have a lot of time to wait for replies in your search to find the perfect contractor, you may want to try an alternate approach.
Post a job on a freelancer site. There are a lot of different websites that serve as brokers and marketplaces for freelancers of all kinds; most of these sites let you post a job opportunity for free. If you put together a good posting, you should be able to get a wide variety of freelancers ready to work, without having to go to any more trouble than making the initial post.
Be sure to define exactly what you want from your freelancers in your job posting, and how much you're willing to pay. That will keep you from having to sort through a bunch of people who aren't what you're looking for. If you're looking for a specific art or design style, mention it, and provide examples. Ask for portfolios. Put a hard limit on how much you're willing to spend, and keep to it. Remember, you might know exactly what you're looking for, but the people responding to your job posting can't read your mind. You can save all concerned parties a lot of headaches if you are clear and thorough when putting your job posting together.
Working With Freelancers
You should have a contract with your freelancers that outlines everything they're doing for you: what the requirements are when they're getting paid, and how often they check in. Make sure you either ask them to send you progress reports on a reasonable schedule (perhaps once per week), or make a point to check in with them yourself. Sometimes things come up and projects get delayed, and rather than telling the client about the problem, a freelancer will go radio silent and hope they can finish the project before the client realises that it's late. Keeping in constant communication can head off issues like that before they occur.
Never pay the full fee in advance. Escrow is your friend, if you can arrange it. If you don't have an escrow service that you like to use, a simple half-now, half-upon-completion arrangement will work. Be aware, however, that things do happen. People get sick, relatives die, hard-drives fail; there are a thousand different reasons a freelancer you hired might not be able to complete their assignment. You need to be comfortable with the fact that any money you send out before the work is finished might end up being for nothing. Ideally, you'd pay the full price upon completion of the project; there are few freelancers who are willing to work like that, however, so you shouldn't count on it.
If you find a reliable freelancer, someone who does the work well and on time, and keeps you informed of the process, then be sure to remember them for next time. Such people should be treasured.
Working with freelancers can be stressful if you've never done it before, or if you don't do your homework on who you're working with. You can make it easier on yourself by remembering to check in with your freelancers if you haven't heard from them in a while, just in case something's gone wrong. Have them keep you updated with sketches and demos to let you know how things are progressing, and always being polite and understanding. Treating people with respect and courtesy goes a long way toward getting good work done.
Somewhere out there, there's someone willing to sell you whatever skill your project needs. All you need to do is find them.