Your Pro plan is held under the email you used at checkout and attaches to your account the first time you sign in with Google using that same email. There is nothing else to do: no license key, no code to enter.
The two steps
Install Amlin. The page you landed on after checkout starts the download; it is also at amlin.ai/download.html.
Open the app and sign in with Google, using your checkout email. The sign-in screen confirms with Pro plan active.
The app still says Free?
Almost always the sign-in used a different Google account than the checkout email. Open Settings → Account & Sync → Sign out, then sign back in with the matching account.
Paid with an email that isn't a Google account you use? Message us from the chat bubble in the corner or email [email protected] and we'll move it over.
Checking your plan
Settings → Account & Sync shows your current plan, your usage, and a Manage billing button for invoices, card changes, and cancellation. See Upgrading, billing, and cancelling.