That the random matrices $X_1X_1'$ and $X_2X_2'$ are iid is a special case of the following general facts:
Fact 1: Suppose that $X_1$ and $X_2$ are independent random variables (r.v.'s) defined on a probability space $(\Omega,\mathcal F,\mathbb P)$ with values in measurable spaces $(S_1,\mathcal S_1)$ and $(S_2,\mathcal S_2)$. For each $i\in\{1,2\}$, let $g_i$ be a measurable map from $(S_i,\mathcal S_i)$ to a measurable space $(T_i,\mathcal T_i)$. Then the r.v.'s $g_1\circ X_1$ and $g_2\circ X_2$, with values in the measurable spaces $(T_1,\mathcal T_1)$ and $(T_2,\mathcal T_2)$, are independent.
Fact 2: Suppose that $X_1$ and $X_2$ are identically distributed r.v.'s defined on a probability space $(\Omega,\mathcal F,\mathbb P)$ with values in a measurable space $(S,\mathcal S)$. Let $g$ be a measurable map from $(S,\mathcal S)$ to a measurable space $(T,\mathcal T)$. Then the r.v.'s $g\circ X_1$ and $g\circ X_2$, with values in the measurable space $(T,\mathcal T)$, are identically distributed.
Proof of Fact 1: Take any $C_1\in\mathcal T_1$ and $C_2\in\mathcal T_2$. Then $$\mathbb P(g_1\circ X_1\in C_1,g_2\circ X_2\in C_2) :=\mathbb P((g_1\circ X_1)^{-1}(C_1)\cap(g_2\circ X_2)^{-1}(C_1)) =\mathbb P(X_1^{-1}(g_1^{-1}(C_1))\cap X_2^{-1}(g_2^{-1}(C_2))) =\mathbb P(X_1\in g_1^{-1}(C_1),X_2\in g_2^{-1}(C_2)) =\mathbb P(X_1\in g_1^{-1}(C_1))\mathbb P(X_2\in g_2^{-1}(C_2)) =\mathbb P(g_1\circ X_1\in C_1) \mathbb P(g_2\circ X_2\in C_2). $$
Proof of Fact 2: Take any $C\in\mathcal T$. Then $$\mathbb P(g\circ X_1\in C) :=\mathbb P((g\circ X_1)^{-1}(C)) =\mathbb P(X_1^{-1}(g^{-1}(C))) =\mathbb P(X_1\in g^{-1}(C)) =\mathbb P(X_2\in g^{-1}(C)) =\mathbb P(X_2^{-1}(g^{-1}(C))) =\mathbb P((g\circ X_2)^{-1}(C)) =\mathbb P(g\circ X_2\in C). $$
