There's a number of ways to date an artefact, but a lot of them depend on the artefact itself: The material it's made of, where it was found, what condition it's in, and so on. In the case of stone tools, you couldn't, for example, use radiocarbon dating or something similar because stone isn't organic (you could use a similar technique to radiocarbon dating, like uranium-lead dating, but that would give you the age of the rock itself, not the tool made from the rock, so it would be nice for a geologist but not for the archaeologist).

However, if the artefact was found in a properly conducted archaeological excavation, then the archaeologists would have recorded the specific layer of soil from which the artefact came (this is called a Context where I work, terminology can differ). The context the stone tool came from might well contain other artefacts or organic material that can be dated, charcoal is a great one, but maybe bones were found in the same layer. That can go a long way toward giving us a date for the artefact, as long as the context is undisturbed (sometimes human, animal, or geological activity can cause the soil to be disturbed and something very old can be removed from its original context and become associated with much younger material).

But let's say the stone tools were found by someone who wasn't doing an archaeological dig, they just wanted to plough their field or something. The artefact might very well be in a style that has been found in archaeological excavations before, so if the artefact is of a known type then we can say it's a Type XYZ, known to date to around 48,000 years ago.

However, maybe it's a new, unknown artefact that doesn't match up exactly with anything known. Then, as you point out, we try and place the artefact in a typology (a seriation of artefacts based on their features) of similar artefacts: maybe this artefact has features associated with an artefact from, say, 42,000 years ago, but not all of the features, however, it has features, that are new (or 'novel') when compared to artefacts from 52,000 years ago. This artefact, then, might be somewhere in between. Obviously this isn't perfect, and if you don't have absolute dates for the other artefacts in the typology, then you have what is known as a "floating typology" - that is, a typological seriation where the ages of the artefacts are known relative to each other but do not have absolute dates. Ideally what we do is more excavation, hoping to find some of these artefacts in contexts that we can date in order to tie down the floating chronology.

Archaeologists have a few ways that we can date things. These fall into two categories; absolute dates, which correspond to dates in our calendars, and relative dates, which describe the age of an object in relation to other objects around it.

The most usual on-site date for an archaeologist is a relative date based off stratigraphy. If I show up to a site where I know nothing about the culture or it’s materials, the first thing I’m going to look for is a way to relatively date everything there in relation to one another. If I find a certain type of artifact at a stratigraphic unit that was deposited below another unit, I know with certainty that that type of artifact is older than the ones above it, because it had to have been deposited there beforehand. Likewise, if I find artifacts in these contexts that I do recognize, and I can use my knowledge of those artifacts to date them relatively to the other culture’s artifacts. So, for example, if I’m digging in Armenia, and know nothing about Armenian pottery, but DO know about Roman pottery, I may find Roman pottery scattered throughout the contexts in Armenia and can use those artifacts to relatively date the local Armenian ones. This, however, doesn’t allow me to say “this context is from 200 AD to 400 AD,” because none of these artifacts have absolute dates yet, and artifacts can be hundreds of years old by the time they are deposited somewhere else.

We can get absolute dates in a number of ways. The one most immediate to consider is radio-carbon dating, which I am not an expert on but understand the basics. In many archaeological contexts you will find charcoal, which contains carbon. Carbon has a half-life, and decays very slowly over time at a normalized rate. Thus, we can see how much the carbon has decayed, and get an estimate on how old it is. And if I find, say, a ceramic lamp alongside a piece of charcoal that’s dated to be 2000 years old, I can say with certainty that that lamp was deposited right around 20 AD. These dates need to be calibrated, because the rates at which carbon deteriorates fluctuates, so always be sure to check whether or not an absolute date is in BCE or calBCE, because these are slightly different dates.

Before radiocarbon dating, as well as other kinds of absolute dating, archaeologists had to be craftier. One of the best tools for absolute dating is writing, which can be used directly or indirectly. Obviously if I find a lead tablet that says “I wrote this on January 30th, 323 BCE (which doesn’t happen because dates were not written that way, but essentially that could happen in a more convoluted way), then we know that that tablet was made then. One of the ways early professional archaeologists got their dates was by using king-lists; records of kings which would include their dates of reign relative to other kings, some of which could be pinned down to dates corresponding to our calendar. These kings would often have artifacts associated with them, which, when found in other contexts, can give us a rough absolute date. For example, if I am working on a Greek Bronze Age site—which I know is Bronze Age because the pottery there is older relatively than the Iron Age stuff, for which there is writing to describe— and I find an Egyptian scarab dating to the reign of Akhenaten, then I know that this site could date to around the period which Akhenaten ruled, because those scarabs are only associated with him. I can’t say for certain a calendar date I am looking at, but I could narrow it down to at least a century, probably less.

In the case of your Sri Lankan artifacts, I would guess they performed some kind of absolute dating, since relative dating wouldn’t give you a precise date like this for something so far in the past. Since they’re dating stone tools, I would guess that they performed potassium-argon dating, which in principle works the same way as radio-carbon, but because potassium has a longer half-life, it can be used to date absolutely an object much older than radio-carbon can date with less precision. I am no expert on dating methods, but this is the basics. There are many other methods of absolute and relative dating, and I encourage you to look into them yourself if a particular one interests you.