"The flame within" is clearly metaphorical language -- of the sort as "the
light in one's eyes" or "fire in one's belly." Unless you look closely at
Gandalf, it seems all you saw was an old man clad in dingy grey; and even when
he was clad in shining white, unless you looked closely, it was still just an
old man you saw.
But, as the Reverend, suggests the passage should probably
be read next to that detailing the encounter with the Balrog, who represents
the "fire that devours and wastes" in opposition to the "flame of Anor."
I'd take "the flame of Anor" to refer to the light of the Sun which nurtures
and warms; this is the "fire" which Gandalf wields against the "flame of
Udun/Hell."
The Balrog, as a fire-demon from Norse mythology, which has
Surt appearing at Ragnarok, is (mythologically) quite literally a
world-destroying, devouring fire. The Gandalf's body should, perhaps
metaphorically, be suffused with the bright light of day, which occasionally
shines forth when you look closely, is only
fair.
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All that is gold does not glitter,
Not all those who wander are lost;
The old that is strong does not wither,
Deep roots are not reached by the frost.
From the ashes a fire shall be woken,
A light from the shadows shall spring;
Renewed shall be blade that was broken,
The crownless again shall be king.
