The best guess at a cause when ArcGIS Explorer desktop fails on first start, especially if ArcGIS Explorer has never ran successfully on the machine before, is the graphics card driver and/or graphics card.
Often getting an updated driver for the graphics card will resolve the issue, but in some cases it can cause an issue. We have had reports where ArcGIS Explorer was working fine, and then one day quit working, the quit working also coincided with machine updates that changed the driver on the graphics card.
If you have an integrated or mobile graphics card (common in laptops and notebooks), look for an updated driver from the computer hardware vendor (HP, Toshiba, Dell, Lenovo, Acer, Asus, etc..) If you can not find a driver from the vendor or the vendors driver does not work, try the graphics card maker.
For those that encounter problems it would be helpful if you listed the Computer, graphics card and driver as well as the experience... Doesn't start, computer background shows where the map should be, 2D works - but 3D does not, etc...
Here is a short list of computers and drivers where ArcGIS Explorer is known to work.
ArcGIS Explorer works with these, and many more, these are machine that I have used ArcGIS Explorer desktop on.
Dell NVIDIA Quadro FX1400...........10/29/2007...........6.14.11.6265
Dell NVIDIA Quadro FX 1800M........2/19/2010.............8.16.11.8921
Dell NVIDIA Quadro FX 1800M........7/31/2010.............9.17.12.5922
Dell NVIDIA Quadro FX3600M.........8/5/2011...............8.17.12.7589
Dell NVIDIA Quadro FX3800M.........8/18/2012.............9.18.13.593
Dell NVIDIA Quadro FX4600............2/9/2012..............8.17.12.9573
Dell NVIDIA Quadro FX4600............12/27/2010...........6.14.12.6645
Dell NVIDIA Quadro FX5500............3/23/2010.............8.17.11.9728
Lenovo Intel(R) HD 4000.................8/21/2012............9.17.10.2843
HP Intel(R) 945 GM/GU...................2/22/2007............7.14.10.1214