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 $$C\cdot D:=\dim\Gamma(C\cap D, \mathcal{O}_{C\cap D})=\dim\Gamma(C, \iota_C^*\mathcal{O}_D).$$  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 $$0\to \mathcal{O}_X\to \mathcal{O}_X(D)\to \mathcal{O}_D\to 0$$ 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.