Why?

Suppose vector ${\bf v}$ has coordinates $\left[{x' \atop y'}\right]_{B'}$ relative to the basis $B' = \{ {\bf u',w'} \}$. This means that $$ {\bf v} = x'{\bf u'} + y'{\bf w'}. $$ Substituting ${\bf u'} = a{\bf u} + b{\bf w}$ and ${\bf w'} = c{\bf u} + d{\bf w}$ into this, \begin{eqnarray*} {\bf v} & = & x'(a{\bf u} + b{\bf w}) + y'(c{\bf u} + d{\bf w}) \\ & = & (ax' + cy'){\bf u} + (bx'+dy'){\bf w}. \end{eqnarray*} That is, \begin{eqnarray*} [{\bf v}]_{B} & = & \left[\begin{array}{c} ax' + cy' \\ bx'+dy' \end{array}\right] \\ & = & \left[\begin{array}{cc} a & c \\ b & d \end{array} \right]\left[\begin{array}{c} x' \\ y' \end{array}\right] \\ & = & \left[\begin{array}{cc} a & c \\ b & d \end{array} \right][{\bf v}]_{B'}. \end{eqnarray*}