The way it works is your Irish Ltd. company, run by a Ryanair approved accountant, will invoice Brookfield or Storm, as applicable. Your Irish Ltd. company pay out your salary which will be deposited into your personal bank account.
The difference to other companies using contractors, e.g. Norwegian, (and please correct me if I am wrong) is that you are hired directly by the Agency providing pilot services. Hence you've got a proper contract, taxes and social security matters sorted (even though tax rates can differ depending on where you are based).
I believe I am speaking for the majority of Ryanair contractors that the biggest issue is the complexity of tax/social insurance affairs and staying within the law. The social contributions are a significant cost to any company that Ryanair has (cleverly) managed to dump on the pilots.
737 CL, you asked which contract is best. I think it's highly subjective. FR contract gives high security, basic salary, taxes paid by company, (slight) pension pot, other allowances. BRK pays best hourly rate, but no guaranteed hours. Tax and social contributions can be quite high depending on your circumstances and you may see more or less take home pay than FR contract. Storm has same structure as BRK. Pays slightly less but has min. 30 hours guaranteed per month.
To mud the waters further, no FR contract is the other alike. They're tailored after living expenses in the country you are based. Also keep an eye on your sector pay if it's referred to as net or gross.