Hi Check Airman,
I'll try to make it short: Crew Agencies can act as intermediary or job placement. Most of the time you will have a contract stating you are not and will never be a company employee, that you are an independant contractor providing a service for a fixed, possibly renewable time period.
You can browse the web for the names you mentioned, you'll find plenty of offer for many airlines and aircraft types around the world. If one in particular attracts your attention, you apply with CV, copies of logbooks, license, medicals, etc. Then you'll be introduced to the airlines if you meet the criteria, and if nobody else with more experience takes the seat.
The crew agencies will take a monthly fee on your pay check for a contract pilot, something like 800 to 1000 USD, depending on their agreements with the airline. For placement, they may take 6K to 15K to place a pilot on a typical 2-3 year contract.
Pros: You can change shop when you please every 2-3 years if you like to visit different countries, or if a contract next door is offering more for the same amount of work! The tax system is mostly left to you, so you are responsible to deal with your declaration of earnings...(wait, is it my country of residence, or country of the airline, or the one of my contractor, or my place of birth...I'm getting confused...
) . The pay is usually good, some perks come in like travel and accomodation in some places.
Cons: Could be tough on the family, ID travel not always included, no seniority, little or no chance of an upgrade or type change, or you could end up in a place you love to hate for a while, but you might want to stick around till the end of your contract if you want another job with the same agency.
I've been on contracts most part of my career, not complaints, discovered many great places and people, and a few not so nice, but for me the balance is really positive so far.
One advice: if you can, get time on the left seat before trying to market yourself. Despite the lack of pilots on the market now, many companies are going full throttle in training their own cadets (new F/Os) and you'd find it hard to compete with a local pilot for any upgrade or benefits.
Hope this helps.