I'd like to expand a bit on the excellent comments of Charles Staats and Donu Arapura. They both suggest understanding the self-intersection number of a curve as the number of fixed points of an infinitesimal deformation of the curve, which is manifestly the degree of the normal bundle when such a deformation exists. Here's a slightly more pedestrian route, which I think has the benefit of being rigorous and almost as intuitive.

Suppose we have two curves in a surface: $$\iota_C: C\hookrightarrow X, \iota_D: D\hookrightarrow X.$$ If $C\cap D$ has dimension zero, the intersection number should manifestly be $$C\cdot D:=\dim\Gamma(C\cap D, \mathcal{O}_{C\cap D})=\dim\Gamma(C, \iota_C^*\mathcal{O}_D)=\dim\Gamma(C, \iota_C^*\mathcal{O}_D(D)).$$ (The twist in the last equality does nothing by our assumption on the dimension of $C\cap D$ --- I've just inserted it to simplify things slightly later on.) We'd like to write this as an Euler characteristic, to make it constant if we vary $C$ or $D$ in a flat family. But this is easy; since $\mathcal{O}_{C\cap D}$ has zero-dimensional support, it has no higher cohomology, so its Euler characteristic equals $C\cdot D$ as defined above. Line bundles are nice (and more importantly, are acyclic with respect to restriction), so we use the short exact sequence $$0\to \mathcal{O}_X\to \mathcal{O}_X(D)\to \mathcal{O}_D(D)\to 0$$ to rewrite this Euler characteristic as $$\chi(\mathcal{O}_X(D)|_C)-\chi(\mathcal{O}_C)=\operatorname{deg}(\mathcal{O}_X(D)|_C).$$

I think this is a reasonably intuitive motivation for the definition of the intersection number. So to fully answer your question, one should give an intuitive reason for why $\mathcal{O}_X(D)|_D$ is $\mathcal{N}_{D/X}.$ Of course, this is just the definition of the normal bundle, but let's motivate the definition. First, why is the conormal bundle $I/I^2$, for $I$ the ideal sheaf of a closed subvariety $V\subset X$? Well, an element of $I/I^2$ is precisely a function on $X$ vanishing at $V$, but ignoring higher-order terms. A section to the normal bundle precisely takes functions $f$ defined in a neighborhood of $Y$ and differentiates them--but the partial derivative should depend only on the first-order part of $f$. So the normal bundle should be precisely $(I/I^2)^\vee$. This is another name for $\mathcal{O}_X(D)|_D.$

I hope that was some reasonable intuition/motivation.

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