In brief: Substance means the bearing element, what underlies everything. Subject in Hegel does not mean „person“ but: something that brings itself forth. The demand is therefore to think what bears not as a resting ground but as a movement that makes itself into what it is. The sentence is the programme of the whole system.
The sentence
It stands in the Preface to the Phenomenology and is Hegel’s own short version of what he wants:
Everything turns, on my view — which must justify itself only through the exposition of the system itself — on grasping and expressing the true not as substance but just as much as subject.
Phenomenology of Spirit, Preface
Note the „just as much“: substance is not abolished. It is not to be replaced but to be thought also as subject.
What it is directed against
Against Spinoza — and against Schelling. Spinoza’s substance is the One that bears everything; every singular thing is only a mode of it. Hegel does not hold this to be false but unfinished: in a mere substance everything determinate disappears without its being said why it appears at all. That is why he calls such systems not pantheisms but „acosmisms“ — the world is lost in them (Encyclopaedia § 573 Remark; see Was Hegel a Christian, a pantheist or an atheist?).
The sharpest formulation is aimed at a contemporary. Whoever passes off his absolute as the simply undifferentiated passes it off
for the night […] in which, as one is wont to say, all cows are black — [this] is the naivety of emptiness in cognition.
Phenomenology of Spirit, Preface
What „subject“ means
Not: an I that contemplates the substance. Rather: that the substance is the movement of distinguishing itself, of remaining with itself in what is distinguished, and of thereby first becoming what it is. Subject in Hegel is a relation, not a thing — the reason why he determines spirit as „being with itself“ and not as a substance that has something.
How this looks in the philosophy of right
Here the sentence becomes testable. Ethical substance is what underlies a society and precedes the individual: custom, institutions, the law in force. § 151 expressly calls habit „a second nature“. Were that all, ethical life would be a compulsion to which one is subject.
The decisive addition stands in § 257: the state is the actuality of the ethical Idea as
ethical spirit — as the revealed, self-transparent, substantial will, which thinks and knows itself and accomplishes what it knows, and insofar as it knows it.
Groundlines, § 257
Substance that thinks and knows itself — that is substance as subject, literally. And § 260 draws the practical consequence: the universal does not hold without the „particular interest, knowledge and willing“ of individuals. An order that is borne without being known is, in Hegel, not yet actual.
Why this is not mere terminology
From the distinction follows the standard of the whole book. An institution is not justified by the fact that it exists and bears — then it would be mere substance. It is justified only when those who live in it can recognize themselves in it. That is why Hegel demands the publicity of the laws (§ 215), the publicity of the administration of justice (§ 224), the publicity of the debates of the estates (§ 314 f.) and freedom of the press — not as concessions, but as that which first makes substance into subject.
Sources. Phenomenology of Spirit, Preface. – Encyclopaedia (1830) §§ 151–159 (relation of substantiality, transition to the concept), §§ 381–384, § 573 Remark. – Science of Logic II, Third Book, First Section. – Groundlines § 144–§ 157, § 215, § 224, § 257, § 260, § 314 f. – Literature. Klaus Düsing, Das Problem der Subjektivität in Hegels Logik, Bonn 1976; Robert B. Pippin, Hegel’s Idealism. The Satisfactions of Self-Consciousness, Cambridge 1989. – In the glossary: ethical substance, spirit, concept.