The product seems to comprise two elements: the API Gateway, and the Dashboard/Portal, which is a web-based management interface for your API Gateways. According to their front page:
The Tyk Open Source API Gateway has no restrictions and is Open Source in the truest sense of the world.
Tyk Dashboard and portal is free for developers, nothing is missing, you have the full-fat product ready for non-commercial use.
The source on their GitHub site is the API Gateway, and it is available under MPLv2; the software they make that requires licence keys is the Dashboard/Portal. This looks to me much like the business model known as Open Core, inasmuch as the core of the product, the gateway, is genuinely free software, but the shiny bits around the outside, the bits that help you manage hundreds of these things in deployment, are non-free (though I note that they are zero-cost in limited form to developers, presumably to help them bring it into an organisation and convince management that it's worth paying for the non-free stuff). You then ask:
What prevents me from removing the license key restriction and build the project?
As far as I can tell, there is no such restriction in the free part; the part that has licence key restrictions is the non-free part, and you are prevented from modifying and copying that by copyright law, just as you would be with any other proprietary piece of key-activated software.
Why this product is not fully available for free?
I'm sorry? If someone comes up to you and offers you a free bicycle, is your first response to complain that she's not giving you her car as well? In this case, the creators have decided to make part of their software available along with the four freedoms. If you can make use of that in your endeavours, good luck to you, and go to it. If you decide that it's too much work to make it do what you want, they give you the option to pay for another product they make that may make your life easier, or you can stop using that piece of free software, and find another or write your own.
What allows open source developers to ask their users to pay?
Leaving aside that Tyk aren't doing that, there is no bar to this in any free software licence; indeed, licences that forbid monetisation are regarded as non-free. I recommend reading Stallman's writing on selling free software.