I think saying that the visible light spectrum itself appears circular is a bit misleading. We simply often choose to represent a wide gamut of colors that humans can perceive in the form of a wheel because such a wheel is an intuitive representation with interesting properties. In particular, for example, the RGB color wheel can help one get an intuitive feel for how humans perceive color.
Humans are trichromats which means that we see color by processing information from three different color channels. Specifically, we have three different types of cone cells, each of which sends a signal to our brain, and those signals are processed by our brain yielding a perceived color. As a result, perceived human color is a function $f(S,M,L)$ of the response values $S$, $M$, and $L$ of the so-called "$S$hort," "$M$edium," and "$L$ong" cones. The cones are given these names because the short cone has a peak response at relatively short wavelengths of incoming light (blue), the medium cone has a peak response at medium wavelengths, etc.
What does this have to do with the color wheel? Well, one way to think of a color wheel is that it's a convenient representation of colors you would see upon combining different amounts of three primary colors additively in much the same way that the human eye generates color perception by combining the outputs of your three cones.
Take the RGB color model for example. Roughly speaking, if you were to shine about the same amounts of blue and red light onto a white screen, then when the resulting light hit your eye, your cones would respond in such a way that the signal sent to your brain would result in seeing magenta. If you were to do the same thing with red and green, you would see yellow. If you look at the RGB color wheel, you can see that the wheel displays this fact nicely because the color you see when you combine other colors is placed in between the colors you are combining.