Let $L_1:K$ and $L_2:K$ be field extensions and $f$ be an $K$-linear mapping between $L_1$ and $L_2$.
It says in my notes that if $k \in K$ and $x \in L_1$ then $f(kx)=k f(x)$ and therefore if $x=1$ then $f(k)=k$.
Where does this final implication involving $x=1$ come from? There’s no requirement that $f(1)=1$ as far as I’m aware.
You are right — there is no such requirement. For instance, if $K = L_1 = L_2$, then the map $x \mapsto 2x$ is $K$-linear, but $f(k) \neq k$.
Perhaps this particular function is more than linear (e.g. a ring homomorphism), or there is a mistake in your notes. (Maybe the author of the notes meant $f(k) = k\cdot f(1)$?) I think that's about all that can be said based on the question...