Go to Help -> Check for Updates. SeaMonkey will check the update server for updates to SeaMonkey.
If an update is available, you'll be given the option to update now, or later. Click "Update SeaMonkey".
The update will be downloaded to your computer. When it is finished, you'll need to restart SeaMonkey to apply the update. You can choose to restart now or later. In this case, click "Restart SeaMonkey".
SeaMonkey will close and you'll see a progress window applying the update.
When it is done, the new version of SeaMonkey will open.