A: The cracks almost certainly were caused by the finish on the doors, perhaps a refinishing that occurred right before you bought the house, given that the doors looked fine then.
Although a casual viewer might look at multi-panel doors like yours and see the rail-and-panel construction as purely decorative, this traditional way of building solid-wood doors is an ingenious solution to one of the vexing properties of natural wood: Even after it has been thoroughly dried, including in a kiln, the wood fibers still expand and contract — forever — when the relative humidity of the air changes. The fibers change only a minute amount in length, but they expand or shrink significantly in width. If a door were a solid piece of wood, or multiple planks edge-glued together, the widthwise changes in the wood fibers would make the door shrink enough to expose gaps on the side edges in dry weather and swell enough to stick in wet weather. But by constructing a frame where the pieces of wood are aligned mostly to take advantage of the lengthwise stability of the fibers, the door stays mostly the same size year-round. The panels are designed to float within grooves in the frame, allowing them to shrink or swell without altering the overall dimensions of the door.
Problems develop, though, if too much finish builds up along the edges of the panels, preventing them from sliding in and out where they meet the frame. When dry air forces a panel to shrink but the panel cannot slide, the wood cracks. This might be even more likely to happen when a house has new occupants who, without being aware of the consequences, keep the air drier by using an air conditioner more or running a furnace without a humidifier that previous occupants used.
Forget about just filling the cracks with some type of wood filler. The panels will still expand and contract, so the filler isn’t likely to stay put. The only long-term solution is to restore the ability of the panels to float and then to glue the cracked pieces back together.
Sometimes door panels fit into a recess and have a piece of molding covering one side of each edge, like a picture frame. If you find that, you can gently pry the molding loose, remove the panel, glue it together and reinstall it, after scraping out whatever gunk is in the recess. Mineral spirits and superfine steel wool, labeled 0000, are good for final cleanup. Be sure to number the molding pieces so you can reinstall them in the same places.
More likely, the panels are tucked into grooves cut into the edges of the frame pieces. There is no way to remove these panels without ruining them. Instead, focus on getting them to slide back and forth in the grooves. Because the last coat of finish probably caused the problem, and because your doors have a clear finish, start by using a razor blade to score the finish where the cracked panels fit into the frame. If this isn’t enough, you will probably need to strip the finish to free the panels. Stripping is almost always needed on painted doors. (In theory, you need to strip finish only where cracked panels meet frame pieces, but you might find it easier to remove all of the finish and start over.)
Once the broken panels slide, use a hot-glue gun to attach wooden blocks to both side edges of each of the panels. If a crack has gunk in it, clean out the space with a strip of sandpaper. Dribble wood glue into the crack, and set a clamp against the blocks to pull the crack closed. Clean up any excess glue before it dries, but leave the clamp in place for 24 hours. Then, with a chisel, remove the blocks; hot glue is brittle, so cleanup should be easy. Touch up or refinish the door as needed. Take care not to let finish pool along the panel edges. You’ve seen the harm that can do!