Suppose $X$ and $Y$ are random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ that have the same distribution. Then $\mathbb{E}[XI_{\{|X| \leq a\}}] = \mathbb{E}[YI_{\{|Y| \leq a\}}]$ for $a \geq 0$.
Attempt:
$$\mathbb{E}[XI_{\{|X| \leq a\}}] = \int_{\{|X| \le a\}}X d\mathbb{P}$$
$$= \int_{-a}^a x \mathbb{P}_X(dx) = \int_{-a}^a x \mathbb{P}_Y(dx) = \mathbb{E}[YI_{\{|Y| \leq a\}}]$$
Is this correct?
Yes, it is correct.
More generally if $X$ and $Y$ have the same distribution then - if $f:\mathbb R\to\mathbb R$ is a Borel measurable function - also $f(X)$ and $f(Y)$ have the same distribution because for every Borel set $B$:$$P(f(X)\in B)=P_X(f^{-1}(B))=P_Y(f^{-1}(B))=P(f(Y)\in B)$$and a direct consequence of that is that $f(X)$ and $f(Y)$ have - if it exists - the same mean.
This can be applied here on the function $f:\mathbb R\to \mathbb R$ that is prescribed by: $$x\mapsto x\mathbf1_{[0,a]}(|x|)$$