It's probably useful to note that fine in this sense does not mean "of superior quality, skill, or appearance". Instead it welds two other meanings together:
2 Very small in size, weight, or thickness
and, in my opinion,
6 Exhibiting careful and delicate artistry
In the main "fine print" refers to the smallness of the type used, but whether by coincidence or sardonic intent the "careful and delicate artistry" certainly applies, as a great deal of careful and delicate lawyerly craft goes into it. Note that I'm not necessarily equating "lawyerly" with "duplicitous" or "evil" here, but many people would. Lawyers have to draw very "fine" distinctions in every contract they write, and the terms of a contract are what you are agreeing to in the example you cite.