They are abandoned, judging by the “a dog is not just for Christmas” ads. I think it’s the well funded charities like Battersea Cats and Dogs Home, who can collect them, and a reasonably strong culture for looking for a dog at these shelters before a breeder.

Another effective method is high rates of neutering for pets. There are geographically segregated areas in the UK that can somewhat confidently say they have no strays.

It can take a while to catch a stray, plenty long enough for them to successfully breed, which just multiplies the problem!

People neuter/spay pets

People will NOT feed strays

People will adopt strays

Local councils (municipalities) collect them and there are a lot of shelters

(there is zero rabies here which helps a lot as strays are not inherently dangerous)

There’s no rabies in any animal that’s likely to run up and bite you. A variant of rabies is still present in bats in the UK.

Such dogs almost always die. I think the frequency of strays may be related to how easy it is for a former pet to survive, and it's presumably harder in colder countries.

It's not a miracle. It's just that they are pretty much immediately reported and captured by the likes of the RSCPA or local councils.