According to the Visitor Analytics installation tutorial for Weebly sites (it can be found here), republishing the site is a required step, after adding the application, since it represents a change in the structure of the site.
After connecting our app on your site, it will redirect you to the app in the same tab.
Now, for having it started, you must go back to your Editor again and Publish the changes to your site. Then, you can go back to the app and check the stats.
If it still does not work, something got cached most probably. Try and clear the cache memory of your browser and Republish the site again.
Can't find an answer to your question?
But we're here to help! Feel free to Contact Us if you can not find the answer on our FAQ page or in the Help Center.
Also, the installation process is different in Weebly, but the app itself is almost the same as the standard application. So you might find the answer to your question by checking the FAQ page of our standard application here.