The theory of spherical harmonics shows that $L^2(B)$ is an $SO(3)$ Hilbert sum of the form

$$L^2(B)= \bigoplus_{n=1}^\infty H_n, $$

where $H_n\subset L^2(B)$ are finite dimensional $SO(3)$-invariant irreducible subspaces. In fact $H_n$ are all the irreducible representations of $SO(3)$ and $H_n\cong H_m$ as $SO(3)$-representations iff $n=m$.

If $K:L^2(B)\to L^2(B) $ is $SO(3)$-equivariant, then it indices $SO(3)$-equivariant maps

$$K_{mn} H_m\to H_n,\;\; H_m\ni h\mapsto P_{H_n}Kh\in H_n, $$

where $P_{H_n}$ is the orthogonal projection onto $H_n$. Since $H_n,H_m$ are irreducible representations we deduce from Schur's Lemma that if $m=n$ then $K_{mn}$ is a multiple of the identity and $K_{nm}=0$ if $m=n$. This proves that $H_n$ is an invariant subspace of $K$ and the restriction of $K$ to $H_n$ is a multiple of the identity. In other words, the functions in $H_n$ are eigenfunctions of $K$.

In the paper you mention at page 345 it is shown that the operator $K$ is selfadjoint and, for every $N\geq 0$, the operator $K$ preserves the space $\newcommand{\eP}{\mathscr{P}}$ $\eP_N$ of polynomials of degree $\leq N$. This space has an $SO(3)$-invariant orthogonal decomposition

$$\eP_N= \bigoplus_{k+2\ell\leq N} r^{2\ell} H_\ell $$

where $H_k$ denotes the space of degree $k$ harmonic homogeneous polynomial. The spaces $H_k$ are irreducible $SO(3)$-representations and Schur Lemma will imply that, for every $k$ and $m$ the spaces

$$\bigoplus_{k=0}^m r^{2\ell} H_k $$

are $K$-invariant since $K$ is $SO(3)$-equivariant.