You get no instructions on the installation process from the web site, confermation email, or Impulse. So it is understandable that you would have problems on the install procedure.
If you haven't done it yet, download and intall the Impulse DRM tool and run it.
Go to the My Games tab and update the SOASE game to the latest version.
At the Impulse little blue logo (top left), right click the mouse and select the add/register new program; and select the first Entrenchment listed. It will prompt for the email address you gave when you purchased the expansion and the serial number shown on the email.
The email address used to purchase the entrenchment expansion and the serial number have to match when you register the new game with Impulse. Note that the purchase email for the game can differ from the purchase email name for the expansion; just use the correct one when registering each product.
Then, in the My Games tab of Impulse you should see the entry for the new expansion pack saying it is ready to install so go ahad and do so.
Stardock really, really needs to write all this up nicely and put it on the web site, purchase confirmation screen, and in the purchase confirmaiton email.