I\'m writing a python regex that looks through a text document for quoted strings (quotes of airline pilots recorded from blackboxes). I started by trying to write a regex with
Read the documentation.  re.findall returns the groups, if there are any.  If you want the entire match you must group it all, or use re.finditer.  See this question.
You need to catch everything with an extra pair of parentheses.
re.findall('(("|\').*?\\2)', page)
You aren't capturing anything except for the quotes, which is what Python is returning.
If you add another group, things work much better:
for quote, match in re.finditer(r'("|\')(.*?)\1', page):
  print match
I prefixed your string literal with an r to make it a raw string, which is useful when you need to use a ton of backslashes (\\1 becomes \1).