$N/F$ is finite galois. $E/F,L/F$ galois and $E,F\subset N$. Then $E\otimes_F L\cong\prod_i EL$ where $i\in I$ with $|I|=\frac{G(N/F)}{G(N/E)G(N/F)}$.
Since $E/F,L/F$ are galois, $G(N/E),G(N/F)$ are normal subgroups of $G(N/F)$.
I can finish the proof easily by directly using simple extension argument. $E=\frac{F[x]}{(f)}=F(a)$ and $L=\frac{F[y]}{g}$. So $E\otimes_F L\cong F(a)\otimes_F\frac{F[x]}{g}\cong\frac{F(a)[y]}{(g)}$. Now from galois correspondence, $g$'s factor will split into product of degree $[L:E\cap L]$ polynomials. Now $G(N/E)G(N/L)$ fixes $E\cap L$. So $|\frac{G(N/F)}{G(N/E)G(N/F)}|$ counts exactly how many irreducible factors $g$ will have in $F(a)[y]$.
$\textbf{Q:}$ I am trying a different proof by group ring method. Let $G_1=G(E/F), G_2=G(L/F)$. $E\cong F[G_1]$ as $F[G_1]$ module but this is also $F$ module isomorphism by $F[G_1]$ linearity. Similarly $L\cong F[G_2]$. So $E\otimes_FL\cong F[G_1]\otimes_FF[G_2]\cong F[G_1\times G_2]$. $F[G(EL/F)]\cong EL$ as $EL/F$ is galois. Now $G(EL/F)\to G_1\times G_2$ is injection. I want to write $G(EL/F)$'s coset in $G_1\times G_2$. However, I cannot directly see correspondence with $\prod_{gG(N/E)G(N/L)}EL$. I want to follow this path. Hint will be sufficient.
Ref: Taylor Frolich Algebraic Number Theory Exercise Chpt 1.2