If a polynomial $P(x)$ (think of it as a function that accepts input $x$ and returns an evaluation result) evaluates to zero for certain input values for $x$ (a very simple example could be $x=x_1, x_2, x_3$), then as stated in the first quoted passage in the question, $P(x)$ will be a multiple of a lowest-degree polynomial $Z(x)$ that is zero across those input values for $x$ (here it would be $Z(x)=(x-x_1)(x-x_2)(x-x_3)$). This means that if $P(x)$ is divided by $Z(x)$, the quotient (call it $H(x)$) will still be a polynomial. The prover computes $H(x)$ by actually performing the division, namely $P(x)/Z(x)$, and this $H(x)$ is sent from the Prover to the Verifier as the “Proof”. The Verifier then confirms that the product of $H(x)$ and $Z(x)$ is indeed equal to $P(x)$, and if so the Verifier accepts the proof. The Verifier needs to also confirm that $H(x)$ is actually a polynomial, and not some other type of function (for example, it should not be a rational function, which is a non-trivial fraction with polynomial numerator and denominator). In practice, this is pretty much confirmed by default due to the way that the Prover communicates or calculates $H(x)$. For example, if the protocol requires that the Prover sends over the coefficients of $H(x)$, then it is obvious that it is a polynomial. Or, if the protocol requires that the Prover computes $H(r)$ for a secret value $r$ (meaning that $H(x)$ is evaluated at $x=r$), then there will be a “Common Reference String” provided to the Prover that includes encodings (e.g., in the exponent of a group generator in a pairing friendly group) of powers of $r$ up to a highest power (e.g., ${g^{r}, g^{r^2}, g^{r^3}, g^{r^4}, g^{r^5}}$), so the Prover is forced to only propose a polynomial $H(x)$ (evaluated at $x=r$) with degree up to that highest power (in this simple example, 5).
