I think it's meant to be clever wordplay – a pun of sorts.
The word raw has several definitions; here are a few:
1 (of food) not cooked ⇒ raw onion
2 (prenominal) in an unfinished, natural, or unrefined state; not treated by manufacturing or other processes ⇒ raw materials for making steel, raw brick
3 (of an edge of material) unhemmed; liable to fray
4 (of the skin, a wound, etc) having the surface exposed or abraded, esp painfully
5 ignorant, inexperienced, or immature ⇒ a raw recruit
Which meaning is being used here? At first glance, the obvious choice is Def. #5: inexperienced. However, if you try inserting the synonym into the original sentence, it's not a very good fit:
A ballet dancer who did not apprentice himself inexperienced would never expect to perform in Carnegie Hall.
The phrase "apprentice himself inexperienced" doesn't make much sense. It is reminscent of phrases like "work himself to the bone," but the word inexperienced doesn't work quite like raw does. However, if we switch to Def. #4, we can insert the synonymous words, and the sentence will still make sense:
A ballet dancer who did not apprentice himself until he feet were blistered and calloused would never expect to perform in Carnegie Hall.
That sentence makes sense; essentially, it means: Unless you work your tail off, you won't make it to the top.
However, the word raw works very well with the word apprentice; together, they form a double-meaning of sorts. When I read the original:
A ballet dancer who did not apprentice himself raw would never expect to perform in Carnegie Hall.
what comes to mind is:
A ballet dancer who, when he was raw (that is, when he was unrefined and inexperienced), never had a mentor who worked him until he was raw (that is, until his feet were blistered and sore) would never expect to perform in Carnegie Hall.
This literary device – deliberately choosing a word that shades a sentence with a double meaning – is neither new or uncommon; Shakespeare is famous for it. Given that the original sentence is found in an essay about great writers, it's not surprising that the author tried one of the tricks of great writers.