Any time you need to make an exponential number of things out of a linear number of parts, you should think "subsets": a set of size $k$ has $2^k$ subsets. In this case, the approach is to find a structure where you have a linear-sized set of choices and each subset of those choices leads to a different cycle.
In Yuval's example, you have a linear number of diamonds and, in each one, you can choose any subset of "left" vertices from the diamonds and put those in your cycle, giving exponentially many options.
In Apass.Jack's example, if you start going clockwise around the outside, you can use any odd-cardinality subset of the the radial edges to move back and forth between the outside and inside before arriving back at your start position. (If you uncrossed the two edges at the top, you'd need an even-cardinality subset of radial edges, instead.)