It merely means that they are most likely looking for spaces with client-side JavaScript, and rejecting any string that contains spaces.
Or, they're checking the string server-side, before hashing and storing the hash to their persistent data store.
Either way, as long as the connection is secure, it's reasonable to inspect the string in plain-text, both client-side and server-side, before creating the hash, as long as they actually create the hash when they do actually decide to save a representation of the password, rather than saving the plain-text password itself.
Part of me would like to guess that the reason spaces might be forbidden in passwords (and pass phrases) would be to prevent frequency analysis. But that doesn't really make much sense, when it comes to hashes, does it? So then, might the case be that they are storing the passwords with reversible encryption, and decrypting the token for a match? In my mind of minds, I doubt it, but you never know. What I really think is that the password policy is simply contrived and idiotic, and whatever rationale they've used to forbid spaces is silly and ill-conceived.
Either way, I agree with the author, whole-heartedly. That is a damned frustrating sign-up process. Two-factor account creation AND a CAPTCHA check? Holy moly!