Afters years of being one of the most quotable political operatives around, I developed a case of political laryngitis.
All quotable's meanings have to do with putting something in between quotation marks, making excerpts, citing and repeating. None of these feel right in the context of this sentence.
To mean, quotable in this sentence probably mean to be trusted. But I do see how the meaning "worth quoting" can fit here, somewhat.
Anyways, I am not quite sure. My command of English is pretty bad.