It can be difficult to tell the difference between rocks that have been modified by people and rocks that have been shaped by natural processes. That being said, there are things to look for.
First is material. From the photographs in the linked article, it appears that the purported tool is made from some kind of fine-grained silicious material (high in silicon, rather than magnesium and iron, as evidenced by the color), whereas the surrounding rock appears to be basalt (mafic, therefor darker in color). If you work in an area, you get to know the geology of the region, and where rocks come from. Seeing rocks far from their sources often indicates human curation. That being said, it seems unlikely to me that anyone would bother to curate a general tool like the ones photographed, so that probably isn't going to be a huge factor in this case.
Second, after seeing hundreds or thousands of stone tools, you get good at identifying them. It is kind of like chicken sexing---it may be difficult to quantify *exactly* why something is a tool, but people get really good at it, none the less. Again, this isn't the whole story, but it gives you an idea about why one might pick up a rock in the field. People who have a lot of experience and training are more likely to recognize potential tools.
Third, there are morphological indications of human modification. Rocks that fall and break naturally tend to have random patterns of flaking, whereas intentionally modified rocks will show flaking that is concentrated in a particular place. This isn't foolproof (indeed, there were purported pre-Clovis tools found in California a few decades ago that, upon closer examination, turned out to be naturally formed), but, again, it is an indication.
Fourth, it is often possible to tell a tool from other contextual clues: is it near a hearth? a pile of animal bones? other easily identified tools? Again, given the age, this is unlikely to be useful in this context, but you asked a more general question, so this is part of a more general answer.
Finally, there are lab tests that can help. One can check for residue (i.e. blood or plant reside that might indicate use in preparing food), or microflaking that might indicate use, for example. These are things that you can't see in the field, and almost certainly can't see in a photograph that was taken in the field.