The story referenced was written to point out how difficult to parse the sentence, and it is written exactly as spoken which makes it more difficult to read. Cleaning up the "errors" in speech, I think it would be written as:
It depends on what the meaning of the word 'is' is. If 'is' means "is and never has been", that is one thing. If it means "there is none", that was a completely true statement.
The second thing that makes this difficult to parse is the mixing of negatives with a seemingly positive sense of "is". This is because the sentence which the "is" came from was "There is nothing going on between us".
Therefore it would probably be easier to parse it as the phrase "is nothing":
It depends on what the meaning of the phrase 'is nothing' is. If 'is nothing' means "is nothing and always has been nothing", that is one thing. If 'is nothing' means "there is currently nothing", that was a completely true statement.
It is basically distinguishing that when he made the remark "There is nothing going on between us", he meant that "As of right now, there is nothing going on between us."
But this leaves room for "something" to have occurred before "right now".
At least according to him.