I ran into the same issue - downloaded and installed Mavericks yesterday, but forgot to save the download first so I could make a USB installer. When I went to re-download, I got the "Mavericks is already installed..." message, clicked Continue, the wheel at the top of the window started spinning, and then...nothing. I let the wheel spin for the better part of an hour several times without seeing any progress. Neither restarting App Store, nor Control-clicking the 'Download' button, nor attempting to download from the 'Purchases' tab succeeded in starting the download.
I decided to try temporarily spoofing my OS X version, and it worked! Here's a step-by-step:
1) If it's currently running, close App Store.
2) In a Finder window, navigate to System/Library/CoreServices (you must have Finder set to display hidden and system files as detailed here: https://discussions.apple.com/thread/4505509).
3) Copy 'SystemVersion.plist' to your Desktop, and then rename the copy to 'SystemVersionNEW.plist'.
4) Using TextEdit, open this copy of the file, and change the system version listed under the 'ProductUserVisibleVersion' and 'ProductVersion' fields to something lower than 10.9 (I used 10.7, but I suspect anything lower than 10.9 would work).
5) Save the file on the Desktop.
6) Copy your newly edited 'SystemVersionNEW.plist' into the System/Library/CoreServices directory.
7) At this point, run App Store. Leave it at its initial welcome screen, which today is showing the big 'OS X Mavericks' banner at the top.
8) Now go back to your Finder window(s), rename the original file to 'SystemVersionOLD.plist', and rename 'SystemVersionNEW.plist' to 'SystemVersion.plist', effectively replacing the original file with your edited version.
* Note here that in order to rename the original 'SystemVersion.plist' file, you will probably have to edit its Sharing and Permissions settings by Control-clicking on it, selecting 'Get Info', scrolling down to the bottom of the info window, clicking the lock icon, entering your password, and then changing all the entries in the 'Sharing and Permissions' window just above the lock icon from 'Read Only' to 'Read & Write'. (Once you finish the process and your download has started, you can go back and change permissions back to 'Read Only'.)
9) Go back to App Store and click on the OS X Mavericks banner (if you don't see the OS X Mavericks banner at the top of the window, just do a search for 'Mavericks' and click on 'OS X Mavericks' when it comes up in the search results). If everything has worked as it did for me, you'll now see a 'Free Download' button, instead of 'Download'. Click it, then click 'Install' and your download should begin.
10) As soon as the download appears on your Launchpad with a progress bar, you can go back to Finder, delete 'SystemVersion.plist", and rename 'SystemVersionOLD.plist' to 'SystemVersion.plist', replacing your edited version with the original.
As always, it's a good idea to be careful when editing system files, and make backups before changing anything.
Note that you must have App Store running already before renaming the original 'SystemVersion.plist' file; if you make the change in step 8 and then try to run App Store, it will throw an error and won't start.