After downloading the client, you'll have to copy over your API key from the site in order to use the automatic translation. First, you can get your API key from the bottom of the Settings page, in the Quota section.
Next, open the config.json file in your ztranslate client folder. It should look like this:
{ "server_host": "ztranslate.net", "default_target": "En", "user_api_key": "", "server_port": 443 }
Copy and paste the API key into the "user_api_key" field like so:
{ "server_host": "ztranslate.net", "default_target": "En", "user_api_key": "RN8UF083ZJBOPVBB4YXO70TNBICWDT3I", "server_port": 443 }
Now, run the ztranslate client. You'll see the quota section in the top right of the window now get updated. If translating the screen still doesn't work, then it could either be a graphics driver issue in the game (if you're seeing a black screen in the ztranslate window) or it could be a key binding issue (the log window mentions an error in key_input.py). The fix for these errors are currently a little more involved, so it would be easiest to join the discord and ask for help. The discord link is on the Community page.
To run a pre-made package though, you don't have to input your API key, or even create an account on the site. For that, you can download the package from the downloads page, download the client, and then load the package into the client. At that point, when you switch to a different window on your computer, the client will try to translate that window's screen and display it in its window. If the package does this automatically by default, you can switch it off by unchecking the "Auto Capture" checkbox, at which point you'll have to press the capture button (usually tilde [~]) to translate the screen.
To create a package yourself, again, it's probably easiest to ask on the discord at the current moment.