A lot of people seem to have problems figuring out how to download the game after paying for it. I present to you step by step instructions on how to get the game along with pictures.
Step 1: Go to the main KSP page and click on the Store link.
Step 2: Login with your store credentials. This will be the email and password you used to create your Store account, not your forum account.
Step 3: Once logged in, click on the profile button.
Step 4: Scroll to the bottom of the page where your transaction is and click on the Download link.
Step 5: Now you should be able to click on either the Windows or Mac version download buttons to start the transfer.
KSP Community Manager