There's a fair amount of science involved, but for simpletons like this reporter, Musk basically says the lithium cobalt oxide cells used in the 787 Dreamliner are packed too close together, so that if one cell catches fire, the entire battery pack may ignite in a chain-reaction type situation, which is never good at 30,000 feet. Musk goes on to point out that the cells used in both Tesla vehicles and SpaceX's space-launch rocket are smaller and separated from one another, so that any potential ignition is contained. Musk says offer to help but has so far been rebuffed.
About 50 Dreamliners were recently grounded because of two incidents, one a fire, involving the battery system. The US Transportation Safety Board and the Federal Aviation Administration are currently looking into the cause of the 787 problems.