Denote by $N_c$ the non-orientable surface with $c$ pairwise disjoint crosscaps. So $N_1$ is the projective plane, $N_2$ is the Klein bottle, etc.
Then the "reason" of Klein bottle's exceptionality in Heawood conjecture is the following: for all $c \geq 3$, the complete graph $K_p$ embeds in $N_c$, where $p$ c$ is the smallest integer which is at least $(p-3)(p-4)/6$. The case $c=2$ is exceptional, in fact $K_7$ does not embed in $N_2$. More details are here.
This result on Klein bottle was proven by Franklin, who also showed that there actually exists in $N_2$ a map with chromatic number $6$, obtained by embedding in the surface the so-called Franklin graph