I understand that for an amateur-built that you are required to build at least 51% of the aircraft yourself, but how can there be kits which appear practically completed, just some quick assembly jobs? Could someone explain how this 51% rule works, and what is this percentage based on?
The requirement which states that at least 51% of the tasks needed to make the aircraft airworthy must be accomplished by amateurs is 14 CFR 21.191(g). Presumably this would exclude finishing items like paint, some instruments, avionics, interiors, etc., as those are items which in most cases are not required for airworthiness.
The FAA has several checklists which inspectors use to evaluate the "major portion" requirement. Homebuilders thoroughly document the construction process (or at least, they should!) in a builder's log; a lot of these tend to turn into massive photoblogs. AC 20-27G contains guidance on what this documentation should look like and contain.
The "two weeks to taxi" builder-assist programs that exist out there generally get the basics working, but they certainly aren't "two weeks to flight". Most manufacturers carefully calculate the amount of work they can offer as a kit, based on the number of tasks remaining for the builder to finish.
I'm not a builder, so I can't say for sure, but I understand most of the advantage to these programs is not that somebody is helping you - it's that they can supply tools, jigs, and a lot of behind-the-scenes support like organizing your workflow and providing answers to questions with no turnaround time. They're not there building the plane with you; they're there to make it possible for you to build the plane more quickly... if that distinction makes sense.
EAA also offers a pretty good FAQ for the 51-percent rule, which I admit to paraphrasing here and there!