You first select the .exe file and when AspeQt prompts you to reboot your Atari simply do so. Reboot here means cold-start, pressing RESET may not always work as the RESET key normally does a warm-start. So to be sure, just turn the power on your Atari OFF and then ON to reboot. Whether you should press the OPTION key while re-booting or not depends on the executable, so to be on the safe side always press the OPTION key for the first few seconds of the boot process. Pressing OPTION disables the BASIC ROM so there is more memory available to the executable on XL/XE systems. On a 400/800/1200XL you don't normally need to press OPTION as those computers don't have built-in BASIC.
Another point to keep in mind that not all .exe files will work with Boot Executable option (most do), when this is the case you can load the .exe file from a DOS menu/command line. Another point to look out for, the "Use high-speed loader" option in the OPTIONS menu may not work for all .exe files, if in doubt disable that option before loading the .exe file. Also if you use a high-speed capable OS/DOS on your Atari (i.e. use an upgrade like Ultimate 1MB) don't activate the "High-speed" option in the OPTIONS menu. It will conflict with the OS/DOS high-speed routines.
If none of the above works for you then there may be some other problem, in this case you need to give me more specific information than "nothing happens", but first try above suggestions and see what happens.