The heart of the matter is that a dash interrupts the sentence, because—see? You can interrupt an incomplete sentence—turning it into an anacoluthon—with a dash, and begin a new sentence, or just add a few words less than a sentence or clause.
You may or may not resume the old sentence after a second, matching dash, turning the part between dashes into parenthesis, just as when you enclose something in brackets, or in commas.
Note that overuse of dashes is not good style; but, used sparingly, they are beyond reproach except in very formal language.
In your case, the phrase in other words is a red herring: it is an ordinary adverbial phrase. If you left it out, your sentence would still work and have the same syntax.