In brief: Not by premises and inferences, not by empirical evidence, not by appeal to authorities. His proof is the course itself: a determination is taken for itself, it is shown that it cannot hold up on its own, and the next is the one that remedies the defect. Whoever holds that to be no proof has an objection — but he must make it at a particular point.
What Hegel says about it himself
In § 2 Remark the answer stands in a single sentence:
In philosophical cognition, by contrast, the necessity of a concept is the main thing, and the process of its having become a result is its proof and deduction.
Groundlines, § 2 Remark
A concept is thus grounded by emerging as the necessary result of what precedes. No more — but no less either.
The procedure
The motor is what Hegel calls determinate negation. The decisive passage stands in the Introduction to the Logic:
Since what results, the negation, is determinate negation, it has a content. It is a new concept, but a higher, richer concept than the preceding one; for it has become richer by the negation or opposite of that concept, and therefore contains it, but also more than it.
Science of Logic I, Introduction
What has been refuted therefore does not disappear but enters into the result. The course is accordingly not a chain of assertions but an accumulation: each stage contains all the earlier ones as its moments.
In the Groundlines this can be checked at every transition. An example: abstract right is set out as self-sufficient (§ 34 ff.); out of itself it generates the semblance of right against right, wrong (§ 82 ff.); punishment restores right, but only externally, through a particular will that is itself not bound (§ 102 f.); therefore right must be willed — morality (§ 104).
And what Hegel expressly does not do
He does not define at the outset. § 2 Remark holds definitions in law to be perilous — „omnis definitio in iure civili periculosa“ — and grounds this on the Roman slave, who could not be brought under any definition of a human being.
He does not appeal to feeling or intuition. The manner of „making natural or intensified feeling, one’s own breast and enthusiasm, the source of right“ he calls „the most convenient of all“ and „at the same time the most unphilosophical“ (§ 2 Remark).
And he proves nothing from history. § 3 Remark draws a sharp line: a legal determination can be shown to be perfectly grounded from its circumstances and „yet be inherently wrongful and irrational“.
Where doubt is justified
At the question whether the transitions really are necessary. They are not logical derivations in the formal sense, and Hegel does not claim that they are. Whether a given step compels or is merely plausible can be decided only in the particular case — and well-disposed readers have concluded at several points that it does not compel. That is the serious objection to Hegel, and it is tied to a place: whoever says it does not follow must say where.
The cheap objection, by contrast, is that there is no structure of argument at all. That one is refutable — see Hegel’s concepts are vague, his sentences confused, and there are no arguments — why should one read this?
Sources. Groundlines § 2 and Remark, § 3 Remark, § 31 and Remark (the method as „the own soul of the content“), § 32, § 34 ff., § 82 ff., § 102–§ 104. – Science of Logic I, Introduction („General concept of logic“); Encyclopaedia (1830) §§ 79–82. – Literature. Robert C. Solomon, „Teaching Hegel“, Teaching Philosophy 2 (1977), on the question whether the transitions compel; Ludwig Siep (ed.), Klassiker Auslegen, Berlin 1997; Kevin Thompson, Hegel’s Theory of Normativity. The Systematic Foundations of the Philosophical Science of Right, Evanston 2019 – the most thorough recent investigation of how the Groundlines ground their propositions and what the presupposition of the concept of right in § 2 means methodologically. – In the glossary: determinate negation, dialectic, the speculative.