What is the significance of ⟨B, s⟩ -> ⟨B', s'⟩ as the initial rule in this question about small-step semantics?
It seems you have some doubts about the first added evaluation rule
$$
\frac
{\langle B, s \rangle \to \!\, \langle B', s' \rangle}
{\langle \mathsf{assert}\ B\ \mathsf{before}\ C, s \rangle \to \!\, \langle \mathsf{assert}\ B'\ \mathsf{before}\ C, s' \rangle}
$$

It might be redundant, but I ought to point out it's one rule: the top part is called a premise of the rule, the bottom part is called a conclusion.

We call rules of this type rules of congruence (and the next two are computation rules). One needs rules of congruence to reduce subparts of a bigger expression or command, i.e. reduce something in its context. Usually, one applies rules of congruence until one can use a computation rule to dispose of some parts of the context. Here, the premise serves you as a means of reducing a Boolean expression $B$ to its value $(\mathsf{True}$ or $\mathsf{False})$.

Let me illustrate this by example: if you have a command $\mathsf{assert}\ \mathsf{True}\ \mathsf{before}\ \mathsf{skip}$, and an initial state $s$, then you reduce it to $\mathsf{skip}$ (without changing $s$) using the second added rule:
$$
\langle \mathsf{assert}\ \mathsf{True}\ \mathsf{before}\ \mathsf{skip}, s\rangle \to \!\, \langle \mathsf{skip}, s \rangle
$$

But what should we do if the command we've got is $\mathsf{assert}\ \mathsf{True \wedge \mathsf{True}}\ \mathsf{before}\ \mathsf{skip}$? There is no computation rules for this, i.e. rules that are able to get rid of the $\mathsf{assert}$... part. Here comes the congruence rule: first you reduce $\mathsf{True \wedge \mathsf{True}}$ to $\mathsf{True}$, using some evaluation rule defined for Booleans (not shown here) -- that's why you need the premise ${\langle B, s \rangle \to \!\, \langle B', s' \rangle}$ -- then you use the 2nd added rule to finish off your computation:

$$
\langle \mathsf{assert}\ \mathsf{True} \wedge \mathsf{True}\  \mathsf{before}\ \mathsf{skip}, s\rangle
\to
\langle \mathsf{assert}\ \mathsf{True}\ \mathsf{before}\ \mathsf{skip}, s\rangle
\to
\!\, \langle \mathsf{skip}, s \rangle
$$