I don't know where the OP found his original quote, but it seems to me there is a lot lacking in that explanation. I've found a slightly more thorough explanation, but a full explantion seems to be somewhat hard to come by on the internet!! I've only ever seen a good explanation of this in a book, and it requires quite a bit more than five lines to explain fully - but the shortest explanation which does a half-way decent job of explaining it follows. This from a FAQ on cosmological questions:
The evidence for an accelerating expansion comes from observations of the brightness of distant supernovae. We observe the redshift of a supernova which tells us by what the factor the Universe has expanded since the supernova exploded. This factor is (1+z), where z is the redshift. But in order to determine the expected brightness of the supernova, we need to know its distance now. If the expansion of the Universe is accelerating due to a cosmological constant, then the expansion was slower in the past, and thus the time required to expand by a given factor is longer, and the distance NOW is larger. But if the expansion is decelerating, it was faster in the past and the distance NOW is smaller. Thus for an accelerating expansion the supernovae at high redshifts will appear to be fainter than they would for a decelerating expansion because their current distances are larger.
The part in bold at the end is what was missing from the OP's original quote. It is the correlation between redshift, apparent brightness and actual distance that is used to determine that the expansion is accelerating, not just the redshift.