So I'm curious as to what, exactly, they're modeling and where the dependent and independent variables lie.

I'll try to oblige, but I doubt I'll have the time for it before Christmass break. The articles from the report rely on quite a few other publications which I feel need to be followed to get a better picture. Until then and with a caveat I've done only a very cursory reading of the topic, here we go:

Their model is, in essence, the first term of a Fourier series of a more general log-periodic function progressing towards a critical point (in this case: crash) describing idealised variance of index changes. From the analysis of said variance, they show how previous crashes happened as a result of a local/short-term bubble burst happening on top of long-term bubble growing. In spirit of "picture is worth a thousand words", the self-similarity/fractal part is best seen on their graphs:

however, it's just a quirk/property of the function they used. Intuitively, I think that everything log-periodic will show those kinds of patterns. As to how valid/grounded it is for their model, that will have to wait. Maths part I can get (it's even somewhat similar to what I used for analysis of phase transitions in glass during the last year internship), but economics isn't my forte.

Here's the summary of mentioned "Physica A" publication, which is far more conservative than reports:

The analysis presented above provides not only further arguments in favour of the existence of the log-periodic component in financial dynamics, self-similarly on various time scales, but also indicates that the corresponding central parameter—the preferred scaling factor—may very well be a constant close to 2. In this way it is possible to obtain a consistent relation between the patterns and it allows more reliable extrapolations into the future. It also allows the log-periodicity to pretend to the status of a law. Of course, on short time scales it is a fragile one, as the real financial market is exposed to many “external” factors, such as unexpected wars or other political events, which may distort its internal hierarchical structure on the organizational as well as on the dynamical level. In this connection it is worth remembering that the functional form of the log-periodic modulation so far is not supplied by theoretical arguments and this opens room for some mathematically unrigorous assignments of patterns, as is often needed in order to properly interpret them. Identifying a hierarchy of time scales and a universal preferred scaling ratio is crucial in this connection and very helpful for real predictions. Strict fitting of the lowest order term in the Fourier expansion of the periodic function in Eq. (3) is typically not an optimal procedure. Here it serves basically as a convenient representation to guide the eye.