You should click the Download Client button, which should download a file called Chronicles_of_Arcadia_Installer.exe.
If it's not saving as an .exe file, try right-clicking on the button and selecting "Save link as..." or something similar (this option might vary depending on your browser).
Don't worry about downloading the launcher separately. If you hover over the launcher link, you'll see a note: "This is just the launcher file, which can be downloaded only in case of a launcher issue, e.g., during its update. Note that the client must be installed; the launcher alone will not download the entire game."