I think of downtime as a great way to help players develop their characters and for me as GM to flesh out the larger world and help the story along.
Character and Party Development
Characters will all have something that provokes reactions, negative or positive, to downtime situations. For example, a town on the border between Dwarven and Orcish territory might be a curiosity to a human character, a source of pride for a Dwarf ("We keep the Orcs at bay so you all can lounge about in peace"), and a nerve-wracking place for a half-Orc. Playing up the aspects of a downtime setting can help players plant their characters more firmly in the game world and feel more immersed in it.
This can extend to little bits of flavor you can build around player character interests. The cleric who loves rye ale drags the other characters to a pub reputed to have the best stuff in town. That part of the cleric's personality is know no longer just some words on paper – all the player characters know that about him, and the group is now gaining more cohesion from exploits other than dungeon delving. If there's a big brawl because the drunk cleric insulted an annoying monk, it might be worth indulging in a bit of cliché in order to give the group that bonding experience.
World and Story Development
The world is always in motion, and all sorts of things are going on around the player characters. Even if they're Really Important People, their stories aren't the only ones of importance. In particular, giving the players a sense of the day-to-day activities in the game world as well as the broad sweep of history can help bind them to the setting. If there's a war in the East, not only does wine cost more, it affects people far away – "Those damned Lizardmen! Not only are they killing folk, I can't buy a good glass of red without selling everything I own to pay for it!"
I find it particularly helpful to sprinkle bits of conversation like this into common downtime experiences like equipment purchasing, getting healed up at the temple, and finding a night's lodgings. Not only does it make these experiences less mechanical, it gives you the opportunity as GM to sprinkle in information, rumor, and disinformation. Provided you do enough of it, players will understand that not every piece of news you relay to them is intended as an adventure hook, and not every bit is reliable. This will help you later when you do want to give them an adventure hook, because you can tie it in to information you provided them during down time, and it will feel more natural than if it came from out of nowhere.
You don't have to put much effort into downtime, as long as you have some quick notes about the locale they're in, what's going on in the larger world, and a handful of NPCs. Ex:
Dust Bog used to be an imperial fort many decades ago, but after the empire fell it slowly became a rundown, dusty trade town, nothing but a stop on the road to the coast. It lies right on the border between dwarven and orcish territory, but never seems to get attacked. Dwarves, orcs, and all others are welcome, but anyone who fights in town for any reason shall not be allowed back, and all members of their extended family will be barred permanently as well.
The dwarves to the east are fighting a prolonged campaign against the lizardmen of the Eastern Marshes, which is driving up the price of wine, dwarf-forged metals, and salt, all of which are imported from dwarven lands. Rumors suggest that the orcs may take advantage of the dwarves' fight with the lizardmen, but any orc will vigorously deny this.
Mert Groevner looks like Rush Limbaugh and owns the biggest inn in town. He runs the bar and has been mixing the wine with grape juice and rubbing alcohol. He pours a couple of glasses of the real wine for a customer before shifting to his nasty cost-saving concoction. He dislikes orcs and half-orcs, will try to get an extra copper out of any guest he can, and is full of rumors. In particular, he might tell the player characters a story that the dwarven prince Boldhammer was abducted not long ago by a lizardman war party.
Jessafaer Groevner is Mert's sister. She's redheaded, short, and freckled. Jessafaer dislikes her brother intensely. She's also the town constable, and she spends a fair amount of time in his inn, trying to make sure he doesn't fleece customers and give the town a bad name. She's not a warrior, but by way of a magical amber amulet, she can telepathically call the five members of the town watch to her aid. Jessafaer thinks Mert's story about Boldhammer's abduction is a lie spread by orcs.
Cux Nettleborn is a thin, tall albino from the distant south. He speaks in a thick accent, is very friendly, and happens to run a small but well-regarded establishment that serves excellent rye ale. The place is packed these days because good wine is costly and good rye ale gets you drunk just as fast. Cux enjoys the company of both dwarves and orcs, and is known as a skilled negotiator. Cux thinks the real Boldhammer story is that he went AWOL on a mission, and the dwarven ruling family is trying to cover it up with this abduction story.
Sometimes downtime can be light and inconsequential, and sometimes it can be more than that. If the players don't know which downtimes will be calm and which will have more excitement, they'll see it as something to look forward to, rather than just a bridge between "real" adventures.