The verification process is a necessary step for your launcher to figure which files to add when downloading the weekly update patch. This process can sometimes take a while depending on your computer.
However, if nothing happens after waiting for more than half an hour, you can try going to Computer > Users > Public > StarStableOnline and deleting the Data.ccx and Data.cch files.
After this, you should be prompted to reinstall the launcher once you open it. Click on Yes and reinstall the launcher. After this, the verification issue should be resolved.