Best of my knowledge these drying cabinets had several forms of heating (stoves, steam, steam and gas, etc..) but none used hot water. In fact do not even think they were connected to the home's main heating supply which back in the late 1800's early 1900's could have been any of several various methods.
Things dried in these contraptions are the same as hung on a line or rack indoors. Stiff, and often lacking the sweetness you get from outdoor line drying. As with modern condenser dryers the purpose was to solve a problem; how to dry laundry independent of outdoor weather and or more quickly than anything else.
Smaller versions of these drying cabinets could be had in Britain and other parts of Europe. They were nothing more than a hamper sized metal container with suspended racks and a heating element at the bottom. If the latter was exposed and or your laundry hung to low there was risk of scorching and or things going on fire.
Yes, the idea was same as hanging laundry in a heated room and using a fan/other method of ventilation. That had been done for years (well without the fans until electricity came along), but caused many problems. First and mainly having to deal with wet dripping laundry hanging all over one's home or kitchen. This often created dame and moist indoor conditions. That plus often one's washing took on a whiff of whatever air was indoors. So if the laundry was hanging in the kitchen or near it would be scented with whatever cooking went on. Nothing like Kipper scented undergarments and linens.
As mentioned upthread until vented tumble dryers became smaller and installed in homes, these drying cabinets were state of the art. Commercial laundries by the early 1900's could get steam heated tumble dryers, but these were large affairs not suitable for even great estate homes.
Commercial laundries grew to prefer tumble dryers where possible for the reasons we do so today; things dry more softer and often smoother than simply hanging and being baked dry.
With all these methods one worry was constant in the days when soaps were the main detergent for wash day. If washing was not totally rinsed free of soap the residue could and often did turn brown when exposed to heat. Normally this would occur when ironing, but using any sort of heated drying (cabinet, tumble or even too close to a fireplace/range) produced the same result. In fact laundry manuals of the period warned that things dried in these cabinets often came out with a yellow cast.