Divine Inspiration is an item in the game. It "allows you to upgrade a building from lvl.9 to lvl.10." One Divine Inspiration item is required for each upgrade - the item will be consumed by the upgrade.
Ways to Get Divine Inspiration
Buy with Gems
Divine Inspiration may be purchased from the Shop tab. Choose the General section, and click on the link in the top right corner to go to page 2 of the tab.
Each Divine Inspiration costs 50 gems. Gems may be purchased using real money at a rate of 10 gems = $1. Each Divine Inspiration will cost $5 this way.
Buy a Chest
On the Shop tab, select the Chest tab and locate the Divine Ascension chest. This chest contains 5 Divine Inspiration items.
Each Divine Ascension chest costs 200 gems, or $20 in real money. Each Divine Inspiration costs $4 this way.
Win a Tournament
The Tournament of Might awards gems to the top 10 players by might on a new Domain. The Tournament also awards a single Divine Inspiration item to each of the players in places 11-40. This Tournament runs once on each new domain, usually very soon after the domain has started. This method will only work for the best players on a new server.
The Tournament of Crests awards gems to the top 5 players who have collected the most Crests during the tournament. This Tournament does not award Divine Inspiration items directly at all. This tournament was released at the end of February, 2010. This method also favors the best players who can repeatedly conquer high-level wilderness spaces to try to win Crests.
Get Lucky With Merlin's Boxes
It is possible to win a Divine Inspiration item from the daily Merlin's Magical Boxes game. The odds of winning are very low, but it has happened for a few people.
In practice, the majority of people who have played this game for over a month, playing Merlin's Magical Boxes twice every day, have not seen even a single Divine Inspiration reward. The odds of winning appear to be around 1 in 600 or 0.167%. Do not count on a steady supply of these items this way.
Just Say No
The reality is, this game makes money by advertisements, and by convincing people to buy their way to level 10 with cash. There is only one significant aspect of the game that would be missed by stopping growth at level 9: Catapults. Building Catapults requires: lvl.10 Barracks (1 DI), lvl.10 Fletching, lvl.10 Geometry, lvl.10 Alchemy Lab (1 DI, to research at lvl.10) and lvl.10 Workshop (1 DI, to research Geometry). That's a total of 3 Divine Inspiration items, or $15 cash, just to build catapults.
Most players do not bother with level 10 construction. It is possible to enjoy the game, and to excel at it, without building a single level 10 building. So just hope you get lucky with Merlin's Boxes, and otherwise don't worry about it.
How to Use Divine Inspiration
Once you get yourself a Divine Inspiration all you have to do is to get your building up to level 9. The moment you are able to build the level 10 version you will realize that the requirement window changes a little bit to show a new requirement. All you have to do now is to get the resources together and press Build. The game will stop you to ask if you are sure you want to spend a DI on the building that you have requested.