Besides the traditional linearization suggested by @MarkL.Stone and @Richard, you might consider using the constraints to obtain a compact linearization. Explicitly, multiply both sides of your second constraint by $x_{j,i}$:
$$\sum_k x_{j,i} y_{k,j} = x_{j,i}$$
Now replace $x_{j,i} y_{kj}$ with $z_{i,j,k}$ and impose an additional constraint to enforce $y_{k,j} = 0 \implies z_{i,j,k} = 0$. The resulting linear formulation is:
\begin{align}
&\text{maximize} &\sum_i\sum_j\sum_k \text{cost}(i,k) z_{i,j,k}\\
&\text{subject to} &\sum_j x_{j,i} &= 1 &&\text{for all $i$}\\
&&\sum_k z_{i,j,k} &= x_{j,i} &&\text{for all $i$ and $j$} \\
&&0 \le z_{i,j,k} &\le y_{k,j} &&\text{for all $i$, $j$, and $k$}
\end{align}