Why ask for GRE score on the form?
It's to make the application form easier to review. That way, all the fields on the application form can be printed in one handy package, which could then be sent to a program chair for easy review. 

No one in the admissions office wants to have to gather and collate transcripts, GRE reports, and other application information into one printout (or file) just to be able to send the package somewhere for review. Moreover, when reviewing, no one wants to shuffle through a bunch of papers just to find the information they are looking for. This way, GRE scores are always right there on the left-hand side, in Box 10.

In a word, it's for the sake of convenience. It's much easier on everyone at the university if you simply type your GRE scores into an application form where you are already typing so much other pertinent information. 

As for why you have to submit the official results, too, that's simply to verify the scores you self-reported.