Your license key lives in your account at artifexrouting.com, under Account → License Keys.
To retrieve it, log in with the email and password you used at purchase, then open the License Keys (or Purchases) area. Each Manifyst purchase lists its key along with its status, expiration date, and the sites it is activated on. Copy the key exactly as shown, with no leading or trailing spaces.
Your key also appears in the purchase-receipt email sent when you bought the plugin, so you can copy it from there if you prefer.
Once you have the key, see Add your license key to enter it in WordPress.
If you cannot log in, use the password-reset link on the sign-in page.