That's a rather poor article.
Insulation, and increased temperatures, do not generate extra water.
However, cavity wall insulation means that the inside surface of walls is warmer. The first effect will be that all the water that soaked into the walls from condensation on the cold surface in the past will now start to dry out, and may lead to steamy windows and other signs of increased humidity. The remedy is to increase ventilation to let that water vapour out. Also, with the inside walls being warmer, instead of the water vapour condensing on the walls, and soaking in, where you might not have noticed it, it will find another cold surface to condense on, often windows or in the cold loft. Invariably, the way to get rid of excess water vapour at low cost, is to increase ventilation (there are also high-cost alternatives).
The two things that will increase humidity when you improve your insulation and draughtproofing, are (1) careless installation of loft insulation which blocks the eaves and prevents airflow and (2) sealing windows to prevent draughts (=airflow=ventilation) which previously let cold dry air in, and took warm moist air out. Additionally, some homes have defective brickwork, render or pointing allowing rainwater into the cavity. Insulation companies can be forced to remove, at their own expense, cavity fill they have installed in defective walls, so they are now mostly very reluctant to take the risk, unless the defects are fixed first.
Old wooden windows, especially sliding sash, and fireplaces with open chimneys, are wonderful in an open house. They do however cause enormous draughts, which ventilate your house and keep it tolerably dry, and intolerably cold. If you cure the draughts, you have to provide a replacement source of ventilation. This can best be provided at source of moisture, so a powerful extractor in the bathroom, especially, and in the kitchen, and use of windows or vents in bedrooms, are required in homes that are draughtproofed to modern standards. Unfortunately there are some people who have an aversion to ventilation, and refuse to open windows or turn on extractors. Very sadly, trickle vents on replacement windows are not obligatory.