Lots of people have answered or tried to answer this question. I thought I would have an attempt myself. It has to do with a feeling of security. A tape square on the ground provides a boundary. It is not a solid boundary. It is a visual boundary. We know that domestic cats like boxes. They like the feeling of being inside a box. They like it I believe because it makes them feel more secure.
A tape square allows a cat to feel more secure because it replicates a box at an emotional level. If this is correct then it might indicate that domestic cats still less secure than we believe they do. If you put a tape square on the ground in your home your cat might walk into it looking for that feeling of security.
If this is an emotional response then this behaviour does tell us something about the domestic cat. Either the cat has an imagination to be able to pretend or fool himself that he is sitting inside a box or a solid boundary or he is simply reacting instinctively to a visual image of a square and not rationalizing the fact that it is not a real object. In other words the cat is unable to realise that the tape square is not a real, solid object. That seems unlikely but it may be the truth. Extending this argument, it could be stated that the domestic cat is programmed from newborn kitten hood to like the feeling of a solid boundary against them when they nestle against their mother. So they search for this solid boundary pressing against their body.
Cats also like or need a home range. This is normally the boundaries of the house if the cat is a full-time indoor cat. I have a sense though that this is not about a cat looking for territory or seeking a miniature home range. This has been suggested by one or two people. It is far more likely to be about finding a secure place, even a place to hide.
Another aspect of this behaviour is that although we see videos of cats walking into tape squares on the ground or even circles nobody as far as I am aware has carried out a study on this in a scientific way. I say this because we don’t know how many cats walk into these tape squares and how many don’t. So if you had a sample of 100 cats it might be the case that only 20 like to walk into the square while the other 80% totally ignore it.
A debunking video!
And I think that is a very good point to make because I suspect too that the study would debunk a lot of this speculation about domestic cats liking tape squares on the ground. If more than half of observed cats ignored it then we can’t say that cats like to do it. We can’t generalize. It may be that cats who feel slightly insecure because they are timid or the home is unfriendly have a tendency to seek a secure place and therefore walk towards the tape square.
The fact that some cats do it does make me think that cat owners should do all they can to encourage their cats to feel secure because with security you receive contentment and calmness. Making a cat feel secure is a very important gift that a cat owner can give to their cat. One way to do this is to make sure a cat is not stressed in a multi-cat home.