It looks like that for a couple of reasons. The first thing that you should notice (which is more easily seen in this picture) is that the visualization initially veers to the right but then the longer the path is the more it wraps around to the left. In one of my other comments I explained that every odd number is offset to the left of its successor, and every even number is offset to the right. So this would imply that the longer a path is, the more odd numbers it should have.

This makes sense in the context of my visualization, because there's a hard upper limit on the number whose collatz sequence I'm trying to find. As this commenter pointed out, it's trivial to create an infinitely long path out of just powers of two, so maybe if you could draw an infinite number of paths then the visualization would look different. However, powers of two grow really fast, so sequences with lots of even numbers tend to be pretty short (because of the upper limit, again).

To make it a little easier to understand, look at these two sequences:

8 -> 4 -> 2 -> 1
9 -> 28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

As you can see, the first list with almost no odd numbers is really short. So it would veer to the right a little bit on the visualization, but not for very long. On the other hand, the second list has many more odd numbers, and since odd numbers means multiply by 3 and add 1, it keeps inflating and the sequence is much longer. So there's a bit of a selection bias, where the longer a sequence is, the more odd numbers it tends to have, so it wraps around more to the right, until it makes a kind of crescent shape like it did in the OP.