What is the proper way to use the phrase 'стало быть'? It seems to be used most often when asking for clarification about the result or outcome of something, meaning maybe something like 'it follows that' or 'that means.' But I see 'значит' used as well in the same context. Are they complete synonyms and can be used interchangeably, or do they differ?
"Стало быть" and "значит" are indeed synonyms, so generally speaking you could use them in the same context:
Это синонимы, значит, они могут заменять друг друга. Это синонимы, стало быть, они могут заменять друг друга.
However, "стало быть" is less formal and to me sounds even a bit archaic; in formal context I would prefer use "значит".
Since you mention "asking for clarification", there is another difference. "Значит" can be used to form a question:
Значит ли это, что они могут заменять друг друга?
You cannot use "стало быть" in this manner.
You would be hard-pressed to find phrases so close in meaning used in perfect accordance to their minute differences.
The [original] meaning of "значит" (which grew from the present tense third-person singular form of the verb "значить", which means "designate" or "show", from the word "знак" = "sign") was "shows" or "means" ('has the meaning of'). That meaning is very close to the verb "означает".
The [original] meaning of "стало быть" can probably be deduced from the words of which it consists, "стало" and "быть", literally, "has become to be".
Accordingly, the difference is that "значит" establishes the indirect connection between events or concepts:
"concept1 значит concept2" =
.. "concept1 shows concept2", or "concept1 denotes concept2"
whereas "стало быть" establishes a more direct connection, transitional or transformational relationship:
"concept1 стало быть concept2" =
.. "concept1 becomes concept2" or "concept1 flows into concept2"
However, in modern speech (and writing) those are freely interchangeable.
There are quite a few Russian variants for then / that means. Be it значит (it signifies), стало быть (it becomes), следовательно (it follows), таким образом (in such a way) etc. all these "nails" are interchangeable until they become their "original" meaning, like verb "значить", noun "образ" etc.
The only point is, in the spoken language people usually choose short words over long ones, so, say, "следовательно" sounds quite bookish. In normal conversation the word "значит" is preferred.
The difference in usage doesn't come from literal meaning of the words here. While both expressions have about the same meaning in modern speech, the usage of "стало быть" из much more limited to colloquial cases (still being used not as often as значит) or to refined style in literature. It is also possible to hear it e.g. as a part of some professor's eхplanation for audience at a lecture on a scientific subject, but you'd never find "стало быть" in a scientific writing ("/a, это/ значит" or "следовательно/поэтому" are used instead).