Previous versions of D&D, especially the first two, have included activities that required a chunk of downtime. Typically these have been magic item creation or researching spells, and when only one character was involved, it gave rise to the question: What does everyone else do? The polite answer was "Wait two months until Falzsa the Mighty has built her +5 sickle of doom." and the impolite one was "Go adventuring. Falzsa's player can run the pet wolf." This system promises to grant a raft of nonadventuring opportunities to those other players. Yes, Falzsa's party can still go adventuring without her, but they'll no longer have the excuse of nothing to do.
I suspect downtime activities will be used sparingly by low-level characters, as they always have some adventure or the other to run off to. High level characters are likely to invest more time in them. Whether players bother to partake will depend largely on how useful those rewards are in the campaign. If adventure isn't just something that happens off in the dungeon, and is liable to come to town as well, the PCs may appreciate having a standing army, acquired knowledge or a friendly thieves' guild. The outline of this system looks great and I'm keenly awaiting to see how they flesh it out.