For example, in the smoothing spline problem, finally we get: $$\hat{y}=S_\lambda y$$ where $S_\lambda$ is the (linear) smoother matrix. But how to interpret it?
Furthermore, ESL has a discussion about the $S_\lambda$. If we do the eigendecomposition of it, we get $$S_\lambda=\sum_{i=1}^N\rho_k(\lambda)u_ku_k^T$$ then $$\hat{y}=S_\lambda y=\sum_{i=1}^Nu_k\rho_k(\lambda)\langle u_k,y\rangle$$ It says that "the smoothing spline operates by decomposing y w.r.t the complete basis ${u_k}$, and differentially shrinking the contributions using $\rho_k(\lambda)$". What does that mean?
It seems you need to brush up the language of linear algebra.
A complete basis for $R^2$ is any basis in which any vector in the space can be written as linear combination of the basis vectors.
It can be spanned by for example the vectors $[1,0]^T$ and $[1,3]^T$
In this basis, the vector $[0,3]^T$ can be written as $-1\cdot [1,0]^T +1\cdot[1,3]^T$ because if you multiply in the scalars and add up element-wise it will equal 0 and 3 in those positions.
If the vectors above were eigenvectors to the smoothing operator then what it would do if applied (multiplied) would be to separately multiply the scalars -1 and +1 above with two different values : the eigenvalues of the operator. And the vector belonging to each eigenvalue would be called an eigenvector.
All of this you can read in a first linear algebra book if you have one.