I think they're claiming that people have enormous amounts of trouble reasoning formally about graphs with 3 or more dimensions. Anecdotally this is bourne out by my experience, where it's easy to read an X-Y plot and speak to how the X value influences the Y value(EG linear, cubic, quartic, grows to infinity in one direction, etc). Add a third or 4th dimension and I start to have a lot of trouble making general statements because I have to juggle the impact of the third dimension on the X-Y plot slices.
Please note that I'm not saying it's impossible for people to intuitively understand 3 or more dimensions. Indeed as you say we do it every day. This is not the same as reasoning about it in a general way, which is much harder.
There’s a confound here between an extra “spatial dimension” (x, y, z) and an extra “feature dimension” (x vs y for plain and star-bellied sneeches). Very few people have trouble with the latter, but the former is tricky to plot and interpret.
Please note that I'm not saying it's impossible for people to intuitively understand 3 or more dimensions. Indeed as you say we do it every day. This is not the same as reasoning about it in a general way, which is much harder.