Factorial equality $\text{}\frac{1\cdot 3\cdot 5\cdots (2k-1)}{{2}^{k}2!}$$\text{}=\frac{(2k)!}{{2}^{k}{2}^{k}k!k!}$

In a generating function identity proof in my textbook there is a step that I can't wrap my head around.

$\frac{1\cdot 3\cdot 5\cdots (2k-1)}{{2}^{k}2!}$

$=\frac{(2k)!}{{2}^{k}{2}^{k}k!k!}$

How does one get from the left side of the equation to the right side? Is there an intuitive explanation as for why this makes sense?

In a generating function identity proof in my textbook there is a step that I can't wrap my head around.

$\frac{1\cdot 3\cdot 5\cdots (2k-1)}{{2}^{k}2!}$

$=\frac{(2k)!}{{2}^{k}{2}^{k}k!k!}$

How does one get from the left side of the equation to the right side? Is there an intuitive explanation as for why this makes sense?