To add the Universe widget to your website follow this guide to create and publish an event using universe.
After you've completed setting up an event navigate to your events:
Then on the left hand side navigate to embeddable widget:
You can then choose whichever widget you'd prefer (Note selecting direct link won't add anything to your website) and you should see something like below:
Select App Integrations
And then select Third-Party Integrations
From here click to add an integration and find Universe in the list:
You should now see a field where you can enter your code from earlier:
Press save, this will take you back to your list of added integrations. Toggling the switch will enable the ability to add the form to your pages.
Navigate to the page you want to add the form to, and click the add a section button. You will be able to see your widget under the category "Section Integrations".
Once you've added it to the page that you want the widget to show up on click publish and it should appear on your live site.