Humans started wearing clothes possibly as far back as 170,000 years ago. For comparison, anatomically modern humans arose at least 200,000 years ago out of some other hominin species, although the dates and specifics are debated.
The evidence for the 170,000-years ago date was obtained not from archaeological evidence (which would not have survived), but from calculations based on lice genetics. I will try to explain it.
First note that there are several species of lice that feed off of people. There are lice that lay eggs in people's clothing and lice that lay eggs in peoples' hair directly. These lice are of different species, but had a common ancestor. By measuring the average number of genetic mutations per generation in each species (the rate of genetic change), as well as the average number of genetic mutations between the two species (the total genetic difference between the two species), we can estimate the number of generations it took for the ancestral louse species to become the two lice species that we see now. Multiplying that by the average lifespan of a louse gives us an estimate for how many years ago the split occurred.
But this split must have occurred after the introduction of clothing, for otherwise there would have been no physical separation of lice populations, and without that, speciation probably wouldn't have occurred. So the dating of louse speciation gives us an estimate for when our ancestors started wearing clothing.
hi! this question may be more up the alley of /r/AskAnthropology or /r/AskScience, but fyi, here are a couple of previous posts that may be of interest