A version of BirdieSync only supports a specific major version of Thunderbird (for instance Thunderbird 68.x). So if you install a version of BirdieSync which is not compatible with the version of Thunderbird, BirdieSync add-on will be automatically disabled by Thunderbird.

You may check how versions of BirdieSync and Thunderbird match in BirdieSync download section.
You may also check BirdieSync release notes in forum.

After a major update of Thunderbird:

If you upgraded your version of Thunderbird, it may happen that your current version of BirdieSync is no longer compatible with this new version, because of changes that would require BirdieSync to be adapted and tested.

Check with BirdieSync menu "Help/Check for update..." or in BirdieSync download section if a new version of BirdieSync is available supporting your new version of Thunderbird. You may then update to a more recent version of BirdieSync.
If no version is available yet, then be patient, a new version should be published very soon. It's generally a matter of a few days, sometimes less. Some adaptations and testing must be performed with the official release. Best is done to deliver you a new version quickly. But if you don't want to wait for the new release of BirdieSync, then you may choose to downgrade your version of Thunderbird.

Note also that if the date of this new version of BirdieSync falls after the 1 year period of free updates of your license, it may be necessary to upgrade your license. You may find more details in this FAQ entry.

After an update of BirdieSync:

It may happen that you updated BirdieSync to a version which only supports a more recent version of Thunderbird. Sometimes Thunderbird hadn't been automatically updated, either because you disabled automatic update, or because Thunderbird is updated gradually over several weeks.

To solve this problem install manually the latest version of Thunderbird, compatible with latest version of BirdieSync:
https://www.thunderbird.net/

Remaining incompatibilities after Thunderbird and BirdieSync versions match:

With recent versions of Thunderbird, it may be necessary to restart Thunderbird once more to clear Thunderbird add-on cache and take into account the new version of BirdieSync add-on. But if the problem persists, please refer to this FAQ entry.