Learn English – What does the quote ‘Those things that hurt, instruct’ mean

Quote and unquote are verbal representations of quotation marks, exactly as you suggest in the beginning of your question. In your first example, they take the place of the quotation marks directly:

Alice said Bob quote intervened unquote.

Alice said Bob “intervened”.

In your second example, they indicate that the following (or occasionally preceding) word or phrase is to be understood as though it's a quote:

Alice said Bob quote unquote intervened.

Alice said Bob “intervened”.

When this occurs, the quoted phrase is probably set off intonationally, so you can hear where the boundaries are despite the word order.

Either version can be used with the same meaning, but I think the latter version is especially associated with so-called “scare quotes”. As Wikipedia says:

When referred to as “scare quotes”, the quotation marks are suggested to imply skepticism of or disagreement with the quoted terminology.

To my ear, this is especially likely when quote unquote precede the “quoted” phrase. It's a verbal signal of irony.

He who has a         WHY  to live 
can bear almost any  HOW (to live)

It's a very colloquial way of saying

He who has a         REASON  to live 
can bear almost any  MANNER  of life.

If you have a reason or purpose in life, you can endure almost any misery.

ADDED, to address orthographic issues raised in the Comments:

I have been unable to find the original edition or a critical edition online; but scholarly references appear to use this:

Hat man sein w a r u m ? des Lebens, so verträgt man sich fast mit jedem w i e ? – Der Mensch strebt nicht nach Glück; nur der Engländer thut das.

There are no quotation marks, but warum? (why?) and wie? (how?) are letterspaced. This is a common emphatic device in German orthography; Bernard Shaw was fond of it, too. Some contemporary writers follow another of Shaw's favorite uses with embedded quotations and capitalise these terms (Warum? Wie?) instead; but in German this marks them as nouns.

A translation which preserves Nietzsche's aggressive colloquialism might be:

If you have your Why? of life, you can put up with just about any How? —Man doesn't strive for happiness; only the Englishman does that.

(The last bit of snark is probably not a nationalist sneer but a joke mocking English philosopher Jeremy Bentham and his ‘felicific calculus’.)