score:1
Construing your question narrowly, as asking what "turned within" and "turned without" mean, the answer is plain. The first sentence of the passage in question is, in the original,
Im Mittelalter lagen die beiden Seiten des Bewusstseins --- nach der Welt hin und nach dem Innern des Menschen selbst --- wie unter einem gemeinsmen Schleier trΓ€umend oder halbwach.
Google Englishes it thus:
In the Middle Ages, the two sides of consciousness lay --- to the world and to the interior of man himself --- as if dreaming under a common veil or half awake.
I would tinker, and say
In the Middle ages, both aspects of human consciousness --- the outward looking one facing the world, and the introspective one --- lay, in effect, under a veil, dreaming or half awake.
Both of which are close enough to the translation in your book.
The metaphor here is that consciousness is an eye, seeing either the outside world or one's inward nature. But a vile veil distorts what it sees, and (B goes on to say) the veil was first lifted in Italy.
The larger question, of what Burkhardt meant, or whether he was right, or why he was sure he was right, or what a well-educated mind should make of all of this, is beyond my ken or care.