A body initially at rest fragments into two pieces of identical...

According to the law of conservation of linear momentum, the total initial momentum of the system is equal to the total final momentum.

Since the body is initially at rest, the initial momentum is zero.

Let the mass of the body be $2m$. It fragments into two pieces of identical mass $m$. Let their velocities be $\vec{v}_1$ and $\vec{v}_2$.

$\vec{P}_i = \vec{P}_f$

$0 = m\vec{v}_1 + m\vec{v}_2$

$\vec{v}_1 = -\vec{v}_2$

The negative sign indicates that the two pieces move in opposite directions. Since the magnitudes are equal, they move with equal speeds.