How true is this today? As true as it was then. Extremely true and important.
The reason is simple: it is a principle of middle-game theory that both sides should strive to achieve a central pawn break. In Queen Pawn games, white has a central pawn in d4, which leaves two pawn breaks for black: e5 or c5.
e5 is generally difficult to achieve because of several reasons. Firstly, white tends to have great control of such square with a knight on f3 or bishop on f4 (or g3 in some lines). On the contrary, white's pieces do not tend to have much control of c5 on this openings.
In simpler words, you really want a pawn break and while e5 is difficult c5 is easy.
As a plus, it is better to trade white's d4 pawn for your c-pawn rather than your e-pawn, the reason being that it is generally preferable to trade a side-pawn for a central-pawn.
(I didn't read the excerpt, so I might repeat his summary.)
The purposes of the c5 break is to destroy white's center, free your pieces, and to ease the game. This can be accomplished by the e5 break, but most lines (back then) had b6 played to develop the bishop, so c5 was easier to force.
There are many top level games where the restricted black queen's bishop is the cause of a lost game. There are some examples of black playing a tough defensive game without any center action, but these are difficult.
The axiom is still true and more useful than many rules-of-thumb. such as passed pawns must be pushed.
To imply that ..c5 is essential in all Queens Pawn openings is certainly an overstatement but it does depend a bit on how you define a Queens Pawn opening. If it is merely that White opens 1.d4 then there are many defences than do not require ..c5, but probably Chernev had in mind the reply 1..d5. It is quite popular these days, and was not popular in Chernevs time, to play the Slav Defence 2.c4 c6 with the intention of later playing ..dxc4 and trying to keep the Pawn with ..b5. At some stage of these lines ..c5 might come into consideration but the situation is far too complex for easy generalizations. Wikipedia has an excellent article on the semi-Slav, which continues 3. Nf3 Nf6 4.Nc3 e6. White can prevent Blacks intentions, for example by exchanging cxd or by protecting with e3. Or he may allow the capture and hope for compensation. I have not looked at Chernev in a long time and I forget what he says about the Slav.