I can't find supporting evidence in a standard grammar, but I'll put down my views.
- 1'. The answer is: because humans have well-developed brains.
uses a quote structure with The answer is instead of the typical He said as the dialogue tag. Some might require an introductory colon (as here) or comma, but zero punctuation is often preferred nowadays where the actual quote is otherwise offset (by inverted commas or italicisation). No offsetting at all, as in the original, looks at best informal, arguably unprofessional. Some might italicise (though capitalisation would probably be eschewed):
- 1''. The answer is because humans have well-developed brains.
Internet examples (in a Google search for "The answer is because") include the use of the offsetting comma, a dash, a colon, a comma with inverted commas and capitalisation, a comma with inverted commas and lower case ....
Some would also prefer a complete sentence rather than a fragment (when not reporting a speaker's exact words), but few get really exercised about the judicious use of fragments nowadays.
- 2'. The answer is that humans have well-developed brains.
uses a report structure in standard fashion, with The answer is that as the introductory clause in place of the prototypical She said that. What follows the complementiser that is unproblematically an independent clause. This looks better than  in formal settings (at least).