How does a woman named Jane Collins walk into a Social Security office with documents proving that she is Jane Collins, and a marriage certificate proving that Jane Collins married John Doe, and walk out with a Social Security card with the name Jane Doe?
Social Security program rules 
lists all the laws and rules this government agency follows.
RM 10212.010 Evidence of a name change on the SSN includes US ceremonial marriage as one of the "events that are a basis for name change".
RM 10212.055 B Evidence of new name. "Bride takes the groom's last name." Section D gives the specific example of Lisa Foster becoming Lisa Lee-Walsh by marrying Bryan Lee-Walsh. It says this is "since this new name can be derived from the prior names shown on the marriage document".
The Social Security Administration's website for citizens has this frequently asked questions item for name changes: FAQ entry
That directs citizens to go to this guidance page and select "Corrected" from the options at the top. That displays this information about name changes:
> ## Name change > > If you need to change your name on your Social Security card, you must show us proof of your legal name change. Documents Social Security may accept to prove a legal name change include: > > * Marriage document;