This quote shows a funny way how mathematicians perceive and describe what they see.
In fact, UV and IR singularities are to a great extent of different nature, but they both are results of a bad formulation of theory. They both are absent in a good formulation.
UV singularities are due to "self-action" contained in the interaction term. This interaction term part is wrong and nobody keeps the UV result intact. Rather, it is discarded (subtracted). Thus, the theory is modified. Is the theory local after renormalization - that's the first question to the authors.
After subtracting wrong self-action contributions, the reminder is "good", but, considered by the perturbation theory, it gives divergent results for low-frequency modes. This result is correct - push a low-frequency oscillator with a strong force, and you will obtain huge oscillations proportional to $1/\omega$.
When you have many modes in a superposition, each "soft oscillator" has large amplitude, but after summation (inclusive picture), they give a finite resulting amplitude. These modes are obligatorily all taken into account, not discarded. Only an inclusive picture is meaningful because the purely elastic cross section is zero and each particular mode diverges. Thus, one needs to sum up soft mode contributions to all orders, and that is equivalent to another initial approximation and another perturbative series, i.e., another formulation of theory.
You see, one has to work hard with the original "local" theory formulation in order to arrive at physical results. Is the resulting theory "local" after renormalization and summation of the soft modes - that's the second question to the authors. There are indications that a correct theory formulation is somewhat "non local" (see a popular explanation here or here).