To linearize that constraint as it is can be hard since it is non-convex. Assuming you still want to do that, you would need to introduce binary variables that allow you characterize the function.
Focusing on a single $j$, let first define $w_j=\sum\limits_{I\in I}a_{i,j} x_{i,j}$, with $w_j\geq 0$ and assume you have a bound on such that $w_j\leq UB_j$. Now let $n$ be the number of pieces (linear inequalities) you want to use to describe $\sqrt{w_j}$, and for each piece, let $m_{k,j}$ and $b_{k,j}$ be the slope and intercept of the $k$th piece of the $j$th constraint for $k=1,\ldots,n$, which are tangent lines of $\theta_j=\sqrt{w_j}$ at (finite) points $w_{k,j}\in[0,UB_j]$ (these are the breakpoints in the $w_j$ space), $k=1,\ldots,n+1$. Since the constraint are not convex, only one piece can be "on" in an optimal solution, hence, let $\lambda_{k,j}\in\{0,1\}$ be a binary variable that is one if the piece is "on" for constraint $j\in J$, zero otherwise. Putting all together,
Choose only one piece for crt $j$: $$\sum\limits_{k=1}^n{\lambda_{k,j}}=1 \quad\forall j\in J$$
$w_j$ need to be in the right interval if you choose piece $k$ $$-M(1-\lambda_{k,j}) + w_{k,j}\le w_j \le w_{k+1,j} + M(1-\lambda_{k,j}) \quad \forall j \in J,\,k=1,\ldots,n$$
Definition of $w_j$: $$w_j = \sum\limits_{I\in I}a_{i,j} x_{i,j} \quad\forall j \in J$$
This is the linearized constraint, where $\theta_j$ is greater or equal to the piece that is selected: $$\theta_j\ge m_{k,j} w_j + b_{k,j} - M(1-\lambda_{k,j}) \quad\forall j\in J,\, k=1,\ldots,n$$
As a side note, you have to choose the breakpoints upfront. A plot of $\theta_j\ge \sqrt{w_j}$ (for a single $j$, this a 2D-plot) can help to clarify the linearization.
If your constraints are convex (e.g., the inequality is $\ge$ or you treat it as an SOCP as described in the answer above), then you could implement Kelley's cutting-plane1 method which is an outer approximation method. Those cuts are not cuts in the integer programming sense, so don't add them as cuts. Rather, in B&B add them as lazy constraints. Alternatively, if the MIP is easy to solve, generate a single (Kelley's) cut at a time an re-optimize.
Reference
[1] Kelley, J. E., Jr. (1960). The Cutting-Plane Method For Solving Convex Programs. Journal of the Society for Industrial and Applied Mathematics. 8(4):703-712.