Given a continuous map $f:X\rightarrow X$ on a metric space $(X,d)$, we characterize topological transitivity for the (set-valued) map $\bar{f}:\mathcal{K}(X)\rightarrow \mathcal{K}(X)$ induced by $f$ on the space $\mathcal{K}(X)$ of compact subsets of $X$, endowed with the Hausdorff distance. More precisely, $\bar{f}$ is transitive if and only if $f$ is weakly mixing. Some consequences are also derived for the dynamics on fractals and for (continuous and) linear maps on infinite-dimensional spaces.