My question refers to the document found here. Specifically page 394 of the book (page 14 of the pdf). Theorem 2.5 on that page refers to "the filtration of $H_{m}(Tot)$ obtained by truncating columns of the double complex".
I have no idea what this phrase is supposed to mean. What exactly does it mean by truncating the columns of the double complex? And how does this result in a filtration of the homology of the total complex?
So if $C$ is a double complex, we have by definition $Tot(C)_i = \bigoplus_{p+q = i} C_{p,q}$.
Now, we can look at the subcomplexe $Tot(C)^{p \geq r}$, verifying $Tot(C)_i = \bigoplus_{p+q = i, p \geq r} C_{p,q}$. If you draw $Tot(C)$ in the $xy$ plane, $Tot(C)^{p \geq r}$ is the subcomplex with zero at every $(p,q)$ with $p < r$, hence the name truncating the columns.
Notice that for each $r$ there are maps $Tot(C)^{p \geq r} \to Tot(C)$. This gives submodules $H_m^{p \leq r}(Tot(C)) \subset H_m(Tot(C))$ and this is how your filtration is defined.