@cheekyprogrammer sent me on the right route.

It appears that you have to add an RLS policy for the database to work. Here is how to do that:

1. Go to Authentication > Policies.
2. Find the table you are working with.
3. On the right, you will see a button that says Create Policy. Press that.
4. On the next screen, you will find multiple actions like INSERT or DELETE.
5. Add the ones you need.

And that's it. Now your project should work properly.