They grow at a base rate any time they are not raided. Step 1: Keep your vills safe.
The rate is decremented if there are issues in the village that matter, so Step 2 is solve issues. You can see if they matter on the mouseover of the Hearths in Encyclopaedia or when in village.
There are Settlement governors with perks that increase Hearth growth. Step 2b (optional) is put one of them in charge of the region.
and, for major growth at a cost: Step 3 ~ Put the settlement controlling the region on Irrigation, and fund a little extra input to the project. (Continuous projects don't seem to use the extra funds up, but it increases the production/construction of a settlement so it increases the project effect too.)
A normal situation would be... unraided? grows as +0.4 / d. Issue? make that only +0.2 /d or so. Good Governor? might make it +0.6 or 0.9 per d. Pour on the project? add anywhere from +0.2 for a small castle's project to a blazing +1.0 or so from a big town project.
Soon your Thousand Hearth village with 120+ militia will send a hundred man market party to town, crushing Bandits that come to try them (and capturing some, eventually recruiting some into the party. I've seen Market parties with -Cavalry- as they beat up some bandits who had captured a caravan, and the Villies recruited the Caravan Guards they rescued.) Fair warning, though. Big Market parties are as slow as Looter parties of the same size... so they are very waddly above about 60 members.
