Answers should offer reasonable support, explaining "how" and "why" wherever possible. What that means changes depending on the claims being made.
If you're drawing your answer strictly from the text (see "close reading"), then your claims should be supported through the text. Claims about the author need to be supported through letters, biographies, historical context, etc.
When talking about religious allegory in "The Chronicles of Narnia," it'd be difficult to provide a good answer without citing both the "Narnia" text and some texts about Lewis's religious views.
Other peoples' opinions, theories, and interpretations are solid additional support for claims about those topics--but they usually are neither necessary nor sufficient. They often make answers better, but they aren't the only way to do so and it's important not to just take such things at face value; a great answer will compare scholarly opinion to the other evidence in the answer and draw conclusions.
The Lord of the Rings has been studied so carefully that almost any question we come up with has been considered already, and discussing those considerations could improve the value of the answer.
There are exceptions where critical context is very important, and the further we get from a purely textual analysis the more likely we are to hit on these exceptions.
An answer which mentions Poe's Philosophy of Composition would be pretty weak without mentioning the common critical opinion that Philosophy is an unreliable source.
Lastly, awesome answers help us track down more detail about the subject. Try to credit your sources and inspirations even if you're not quoting them directly.
When writing an answer that draws on Harold Bloom's influence theory I should probably mention where I'm getting the ideas so readers can research it further themselves, even if influence theory is tangential to the question.