How does the "invite friends and earn cash" functionality work?

I'm seeing new startups use the "invite friends and earn cash" invite functionality. How can I manage my financial status with this process? Is it an investment in the short term or long term? Is there any framework for this process or any guide?

This is called "affiliate marketing". You can have a look at Wikipedia: http://en.wikipedia.org/wiki/Affiliate_marketing.

The most common is to pay your affiliates a certain percentage of what you earned thanks to their recommendations.

Depending on where you live there are different legal methods to pay your affiliates. In Poland for example, where I live, you would normally sign a so-called "contract for a specific work", which also obliges you to pay 18% of tax.

In England, I believe a freelancer would have to register as self-employed, but I'm not 100% sure. It would be great if someone could continue my answer and let us know how this works in other countries.

There are online companies who can run an affiliate programme for you. Try to google it. When I google "affiliate programmes" I get lots of results, but they all are from my country, which would be no help for you.