By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The monotonicity lemma now prevents the sequence of rescaled holomorphic discs from "thinning" out and having noncompact image. That is, our maps must use up at least a certain amount of energy for every ball whose center it passes through. The version of the monotonicity lemma used here is not just for compact manifolds such as $(\Sigma',\omega)$ but also symplectizations such as $(\partial\Sigma'\times[0,\infty),d\theta\wedge ds)$, see Proposition 2.71 of Abbas' book "An introduction to compactness results in symplectic field theory".
Restating: The boundedness assumption means precisely that our bubble points are stuck in $\Sigma'$ union a collar $\partial\Sigma'\times[0,k]$ and cannot escape along $\partial\Sigma'\times[0,\infty)$, otherwise you're in Case#1 (in the paper) which has the points running away along the neck of $\Sigma^{R_i}$. As for the rest of the images of the maps, given balls centered around various points in your maps' image, monotonicity prevents the maps from having arbitrarily small area in any of these balls and hence the images of the maps are contained in some overall compact region.
Note: In both the case at hand (Case#2 in the paper) and Case#1 in the paper, our sequence of rescaled maps have domains being open disks centered around the bubble points. It can happen in Case#1 that these disks just shift along with the bubble points down the infinite neck region, which doesn't happen in Case#2.