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You must restrict to the case of blowing up at a fixed point of the torus action, otherwise the manifold is no longer toric. Naively one just replaces the corner with a $\mathbb{P}^1$. On the other hand, the exceptional curve is always a -1 curve, so locally the symplectic manifold looks like the total space $M$ of $\mathscr{O}(-1)\rightarrow\mathbb{P}^1$, which is also a toric symplectic manifold, and its moment polytope $P_M$ can be described explicitly. Thus the moment polytope of your toric symplectic manifold after blow up looks like $P_M$, which explains the reason.