From within the Client, click the small X in the upper right corner of the ad pane (bottom left).
Once clicked, the main Client panel will display information about Ad-Free. To purchase, click the 'Purchase Now' button at the bottom of the page. From here, the main Client panel will display a form for entering your payment information.
Once payment information is entered, click 'Purchase'. Your Client will automatically update and ads will no longer be visible.
If you encounter issues with purchasing Ad-free, or if your client isn't updating, please contact Customer Support.