What is meant by a language have a "finite specification"?
A specification in this sense is any description of the language that allows you to determine exactly what strings are in the language and what strings are not. We're usually quite vague about what exactly constitutes a specification since, ultimately, it doesn't matter. A specification must be describable by some finite string of characters, whether those characters represent English words ("The set of all strings consisting of some number of $a$s followed by the same number of $b$s"), mathematics ("$\{a^nb^n\mid n\geq 0\}$") or somethign else. As your notes go on to say, whatever reasonable meaning you give to "finite specification", there will only be countably many of them, whereas there are uncountably many languages.