Historically boiling was part of the wash day when laundry was done mainly by hand. It was used mainly on whites and colourfast items and followed soaking, soaping, and rinsing.
Boiling served several purposes:
It opened textile fibers to allow any trapped dirt, soil and more importantly soap from the previous soaking and soaping to be released from textiles. Subsequent rinses in hot water carried the muck away.
Boiling destroyed vermin (lice, fleas, bedbugs, etc..) that were all to common then.
It helped whiten and keep whites bright, especially after oxygen bleach (sodium perborate) was discovered and eventually made it's way into the first laundry "detergent" Persil.
Boiling also provided a measure of disenfection in times before modern antibiotics when simple infections could and did kill, especially infants and children.
And so and so forth.
On this side of the pond routine boiling of laundry started to wane as mechanical washing machines became common, and more homes had plummed hot water. Long as one had access to hot water that was >140F to about 160F that was fine to activate the soap which was used for cleaning and subsequent hot rinses would carry away the loosened muck.
If we remember the five main basics of good laundering practice (water amount, time, water temperature, mechanical action, and chemical action), we know the increase in one affects a decrease in others and vice versa. Boiling had the affect of lossening soils without all that beating, scrubbing and other harsh hand washing techniques. However with the advent washing machines to provide mechanical action (usually far more gentle and what was done by hand),hot water temperture could decrease.
American housewives, commercial laundries and so forth also took to and still vastly prefer chlorine bleach for stain removal/fabric whitening. LCB does not require "boiling" wash temps and does it's work quite well in normal household hot water (120F to 140F), so again boiling just wasn't necesssary.
OTHO in much of the UK and Europe boiling held sway and as early washing machines began to be introduced housewives and other consumers made it cleart they wouldn't choose a washer that couldn't do a boil wash.
Early Miele and other washing machines simply had a fire box under the wash tub. Later after electric power became widespread and technology advanced washing machines were built with internal electric heaters.
Part of Europe's love of boil washes has to do with Persil, the first oxygen based laundry "detergent", and the aversion on the other side of the pond to chlorine bleach for laundry use.
Sodium perborate needs water temps at or >140F to really become active. However starting the wash at those temps not only shocks textiles but sets some stains and soils. Hence the famous "profile" wash. Cycle starts with cold water (to prevent setting of stains), moves on to warm (better soil removal and once enzymes were introduced allowed them to work), then finally hot to boiling (to activate the oxygen bleaching agents for whitening, stain removal, etc..).