First of all, both your question and the discussion in the lecture notes, are more general than the chiral anomaly. They apply to all 't Hooft anomalies$^{(*)}$.
The claim is not that the anomaly is an IR effect. In fact if you carry on reading the lecture notes you will arrive at the 't Hoot anomaly matching conditions, which guarantee that the anomaly is an invariant of the RG flow.
The claim is that the anomaly controls the behaviour of the IR of the theory. In fact it tells you that the IR cannot be trivially gapped due to the anomaly. It must contain some interesting physics. Either some topological degrees of freedom, or a gapless phase, or the (anomalous) symmetry should have broken spontaneously at some point before reaching the deep IR.
The most elegant way to see this is through anomaly inflow, where a $d$-dimensional anomalous QFT$_d$ cannot be consistently defined in $d$ dimensions, but rather must be accompanied by a Symmetry Protected Topological phase in one higher dimension (SPT$_{d+1}$) carrying edge modes which exactly compensate for the anomaly.
Now, suppose that somewhere you calculated the anomaly. Equivalently you found the non-trivial SPT$_{d+1}$ phase which captures the anomaly. Now keeping in mind the anomaly matching conditions, you flow with the RG all the way to the IR. With you flows also the SPT$_{d+1}$ phase. Any non-trivial SPT$_{d+1}$ phase can't have a unique ground state. Translated back to QFT this means that the IR of your QFT can't be trivially gapped.
These said, I must mention that I too find these arguments confusing and unsatisfactory. Also these notes are from 2003. Since then there has been tremendous progress on the understanding of anomalies, so these notes are severely outdated in my opinion.
$^{(*)}$ strictly speaking the chiral anomaly as discussed in these lecture notes is not an 't Hooft anomaly but rather an ABJ anomaly. It is, however an 't Hooft anomaly if you consider the gauge field non-dynamical, i.e. just forget about $\int \mathrm{D}A$
