I don't think that what I will now write is the best or quickest answer to your question, but maybe it is worthwhile to set it down because the point of view is a bit of useful folklore.

First let me translate the problem from Morse theory to Hamiltonian dynamics:

**1.** Instead of the submanifold $L$ consider its conormal bundle $\nu(L) \subset T^* M$ defined as the set of all covectors bases at some point of $L$ that vanish on the tangent space to $L$. The conormal bundle is a Lagrangian submanifold.

**2.** Consider the geodesic flow $\phi_t : T^*M \setminus O \rightarrow T^*M \setminus O$ on the slit cotangent bundle. Consider also the flow
$$
D\phi_t : T(T^*M \setminus O) \rightarrow T(T^*M \setminus O)
$$
obtained as the differential of the geodesic flow.

**3.** Last, but not least, consider the bundle of  tangent Lagrangian planes over
$T^*M \setminus O$ that I'll denote by $\lambda(T^*M \setminus O)$. 

The flow $D\phi_t$ induces a flow on $\lambda(T^*M \setminus O)$ that I'll denote by the same symbol. This flow has an important "twist" condition: if $p_x$ is a point in $T^* M \setminus O$ and $\phi_t(p_x)$ is the integral curve passing through it, consider the vertical Lagrangian tangent plane  $V_{\phi_t(p_x)} \subset T_{\phi_t(p_x)}(T^* M \setminus 0)$ and the curve of Lagrangian planes in the (symplectic) vector space $T_{p_x} (T^*M \setminus 0)$
defined by 
$$
t \mapsto D\phi_{-t}(V_{\phi_t(p_x)}) =:\Lambda(t).
$$

**4.** The **twist condition** satisfied by geodesic flows for Riemannian and Finsler metrics (among other non-degenerate Hamiltonians) implies that if $t$ and $t'$
are sufficiently close, then $\Lambda(t)$ and $\Lambda(t')$  are transverse.

**5.** Now we have to translate focal points into this language: Consider $p_x$ on the conormal bundle of $L$ (at the point $x$) and the orbit $\phi_t(p_x)$ $(t \in \mathbb{R})$.

**Proposition.** *The point $y$ obtained by projecting $\phi_s(p_x)$ onto $M$ is a focal point if the Lagrangian plane $\Lambda(s) \subset T_{p_x}(T^*M \setminus O)$
**does not** intersect the (Lagrangian) tangent plane of the conormal bundle $\nu(L)$ transversely at this point. The dimension of the intersection is the multiplicity of the focal point.*

The twist condition immediately gives you that the set of focal points along the geodesic obtained by projective $\phi_t(p_x)$ on $M$ is discrete.