Why are we interested in conformal field theories (CFTs)? One reason is it appears that all quantum field theories (QFTs) can be viewed as points along a “path” between a high-energy CFT to a low-energy CFT (c.f. the roadmap analogy in the article). Therefore, studying CFTs lets us study the space of well-defined QFTs, which are the mathematical framework in which to construct models in particle physics.
At the same time, we can describe seemingly unrelated theories like the 3d Ising model as CFTs. The conformal bootstrap approach has enabled researchers to recently calculate the critical exponents for the 3d Ising model, which effectively “solves” the theory as the critical exponents are universal and characterize all the physical properties of a theory.
The whole business about the “unknown polyhedron” and mapping “the geometry of theory space” means figuring out where these theories (e.g. Ising and other thing) live on this (higher-dimensional) map in relation to each other and potentially how to get from one to the other.
The article briefly talks about the AdS/CFT or holographic principle and how we can use CFTs to study gravity (this is more what I'm interested in) but this is a whole other can of worms. Needless to say, it's also super exciting! Two examples of the types of 'big' questions being looked at are gravity as 'emerging' from a CFT and probing the black hole information paradox / beyond the horizon.
