In page 4, part 2 of assumption I in On the Regret Minimization of Nonconvex Online Gradient Ascent for Online PCA, the papers says "the support of $D$ is contained in a Euclidean ball of radius $R$ centered at the origin, i.e.," $$ \sup_{\textbf{q} \in \text{support}(D)} \|\textbf{q}\|_2 \leq R $$
where $D$ is a distribution and $\textbf{q}$'s are sampled i.i.d from $D$.
What is the meaning of that assumption? Could you elaborate it intuitively for me? What is the significance of such an assumption?
Roughly speaking, the support of a distribution is a closed set that contains [as subsets] all the sets with positive probability. For example, the uniform distribution on $[0,1]$ has support $[0,1]$, the exponential distribution has support $[0, \infty)$, and the Gaussian distribution has support $\mathbb{R}$. See the Wikipedia link for a more rigorous definition.
So $\text{support}(D)$ is a set, and the statement is simply saying $\text{support}(D) \subseteq B_R$ where $B_R$ is the Euclidean ball of radius $R$ centered at the origin. The inequality written in your question is equivalent: it simply states that all elements in the set $\text{support}(D)$ have Euclidean norm $\le R$.
Long story short, the assumption is saying that the distribution $D$ gives zero probability to subsets lying outside of the Euclidean ball of radius $R$.