Treasure chests are like the game's happy little surprises. Sometimes they're given to you, sometimes you uncover them as you claim pieces of land, but they basically represent happy little freebies placed there for you by the developers. Note that you can't always see all of the treasure chests available on a plot of land until you clear it of trees. So if you prefer to keep some of your trees, just know that you may be missing a chest or two. Not a huge deal, but something to keep in mind.
You get your first treasure chest immediately after rebuilding the Royal Vault. It contains 50 crowns and 600 royal coin. After that, every day you'll get a new treasure chest in front of your vault, about 24 hours after you opened the previous day's chest.
One other thing; if you join during a Seasonal Event, the treasure chests may look like something else. For example, during the 2014 Winter Seasonal Event, they all were given the appearance of gift boxes tied up with shiny gold ribbons, like so: