We define a compressing disk of the knot $K$ (a smooth emmbedding from $S^1$ to $S^3$) to be a smooth map $f: D → S^3$ such that $f|∂D = K$ and such that $f| int(D)$ is transverse to $K$. Then $f| int(D)$ has only finitely many intersections with the knot $K$ .
Here: "$f| int(D)$ is transverse to $K$" means that $$ \forall p \in f^{-1}(K), f_*T_p(intD)+T_{f(p)}K=T_{f(p)}S^3 $$
So the Transversality theorem tells us that the intersetions $\{p\in (int D)| f(p)\in ImK\}$ is a 0-manifold. But why it's finite ?