For me, getting into it was easy. I packed off a resume to several of the local contracting companies. They pass the resumes to the ultimate company, who will do an interview (usually very low stress because it's easy to get rid of you) and then the contracting company and the employer work out all the details. (It may have changed, but at the time of my last such deal about 2005, the contracting company absorbed about 15% of what the end company was paying for you.)
Do realize that you need to keep at least 6-9 months of your income in a readily accessible account (savings, money market, or (whoo, I'm old) CD). I never went for more than a week before I could another job, but I have known people who had more trouble.
I have since gone into federal government contracting, which is another whole bag of stinky fish heads.
Essentially since you are contracting, you are outside of the "bubble" where decisions regarding business get made. So while you may have important work on the project, you are not there when they are deciding on business aspects of it, you are usually not invited to any type of sales meetings or meetings with the clients. This is reserved for company people.
Another aspect you can do is that you can outright reject that type of work or slowly move away toward a client that is not requiring that type of work. When you are working in a company, you usually do not have power to do that. Your only option in a company is usually to quit and then hope next company doesn't pull you into that meatgrinder.
The down side is that you don't have any direct influence on those business decisions. (At one point, I wrote about three different login/single-sign-on clients for a web app infrastructure (including Kerberos/Active Directory) before they finally decided that SAML2 was politically/technically the right choice. Frustrating.) You do have indirect influence if you have a good relationship with your team lead, who will likely be a "real person", a real employee.
If you are thinking of contracting/consulting, I'd highly recommend joining a diversified team in the 10-100 range so you can do the parts of the work you love and rely on your team for the rest.
In my case, it was usually a contracting company with 10-100 contractors working for one or more client companies, but the contractor had 0 day-to-day influence on the work I did. (I did try to pick up my paycheck in person and chat with the office staff.)
Might be time for me to revisit that, this being the year when I really think I need to start something new. I have no idea how to make that switch though.
Are you contracting through your own business, or someone else's?