We have $[L:F]=[L:K][K:F]$ if each term is finite.
I claim that if $[L:K]=\infty$ then $[L:F] =\infty$. But I wonder if it can also be concluded that $[K:F]=\infty$.
Proof of the claim: Suppose $[L:F]=r$, finite. Since $[L:K]=\infty$, there exist $r+1$ vectors in $L$ which are linearly independent over $K$. These vectors must be linearly dependent over $F$ (because $[L:F]=r$). But since $F\subset K$, we get a contradiction.
But I don't understand how to prove $[K:F]=\infty$.
Background: The book that I am studying this from first remarks that if $[L:K]=\infty$, then $[L:F]=\infty$. I tried to prove that as above. The book uses a slightly different proof. Then, the book says: 'similarly, $[K:F]$ is infinite.' I tried to repeat the same proof I wrote above to prove this but it didn't work. It's a typo as the answer below shows. But the spirit of that probably is the following: If $[L:K]$ or $[K:F]$ is infinite, then $[L:F]$ is infinite.
This is not true. Take for example $F=\mathbb Q$, $K=\mathbb Q[\sqrt 2]$ and $L=\mathbb R$.