In learning about the celebrated Decomposition Theorem I have found that I am having trouble applying it in even the simplest situations. In particular, I'm considering an example where $f:X\to Y$ is a resolution of singularities of surfaces, say with $Y$ having only one singular point and $f^{-1}(y)$ is a finite union of curves.
Since $X$ is nonsingular it is clear that $IC_X = \mathbb{Q}_X[2]$, but I am unsure where to go next. This example shows up on page 16 in the notes of de Cataldo and Migliorini (https://arxiv.org/pdf/0712.0349.pdf) where they claim that the Decomposition Theorem yields $Rf_*\mathbb{Q}_X[2] \cong IC_Y\oplus T$, where $T$ is the scyscraper sheaf at $y$ with stalk $H_2(f^{-1}(y)).$ I suppose this statement on the stalk comes from the proper base change theorem, but I don't see how this whole statement follows. I believe it is saying that there is only one nonzero term in the decomposition, so my questions are the following:
For which $i$ is the summand of the decomposition theorem nonzero, why is it the only one, and why is ${}^{\mathfrak{p}}\mathcal{H}^i(Rf_*\mathbb{Q}_X[2])[i] \cong IC_Y\oplus T$?
Note: A very similar idea shows up in a mathoverflow post here: (https://mathoverflow.net/questions/18301/decomposition-theorem-and-blow-ups?rq=1) in the context of a more complicated problem which I will tackle understanding at a later date. To paraphrase, they say that by base change, the fact that $f$ is an isomorphism away from $Y\setminus\{y\}$ implies that $Rf_*\mathbb{Q}[2]|_{Y\setminus\{y\}}=IC_{Y\setminus\{y\}}$ (which I certainly agree with) and so ${}^{\mathfrak{p}}\mathcal{H}^0(Rf_*\mathbb{Q}_X[2]) \cong IC_Y\oplus T$. This provides an answer to my first question but doesn't provide me with any additional understanding.