We will construct a bijection
$$
\textrm{Hom}_L(L[x_1, x_2, \dots, x_k], A \times_K L) \ra
\textrm{Hom}_K(K[y_{11}, y_{12}, \dots, y_{1n}, \dots y_{kn}], A),
$$
which is functorial in $A$. To construct this bijection, let $e_1, \dots, e_n$ be a
$K$-basis for $L$. Then
$$
A \otimes_K L = \bigoplus_{i = 1}^k (A \otimes_K K)e_i.
$$
Let $\sigma' : L[x_1, \dots, x_k] \ra A \otimes_K L$ be a homomorphism of $L$-algebras.
We can use $\sigma'$ to define a $K$-algebra homomorphism
$\sigma : K[y_{11}, \dots, y_{kn}] \ra A$. Indeed,
$$
\sigma'(x_i) = \sum_{j = 1}^n \sigma(y_{ij}) \otimes e_i.
$$
Conversely, any $\sigma$ can be used to define unique $\sigma'$. Thus, we have the bijection, so
the $K$-scheme $X := \Spc K[y_{11}, \dots, y_{kn}]$ represents the functor Res$_{L/K}(X')$.
Finally, if $X'$ is smooth, then *** CITE *** implies that $X$ is smooth, so we are done.