= Solution
Let $H=V(L)$ for a nonzero homogeneous linear polynomial $L$. Because $H$ does not contain $X$, the restriction $L|_X$ is a nonzero section of $\mathcal O_X(1)$. Its zero divisor is the <hyperplane-section divisor of a projective plane curve>
$$
[X\cap H]=\sum_{p\in X\cap H}I_p(X,H)p,
$$
where $I_p(X,H)$ is the local <intersection multiplicity>; equivalently, it is the order of vanishing of $L|_X$ at $p$.
The curves $X$ and $H$ have no common component and have degrees $d$ and $1$. The <Bézout theorem> therefore gives
$$
\boxed{\deg[X\cap H]=\sum_{p\in X\cap H}I_p(X,H)=d.}
$$
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