When upgrading from a prior version to Mac OS Big Sur (11.0) or newer you will need to accept a new System Extension to allow the AnyConnect VPN client to function.
Immediately after upgrading the operating system you may see prompts similar to the following:
Click on the Open Security Preferences or Open Preferences buttons to open the Security & Privacy System Preferences window.
At the bottom left of this window click the padlock icon to unlock settings and enter your Mac OS username and password when prompted. Next click on the Allow button next to the mention of "Cisco AnyConnect Socket Filter".
Next you will be prompted to allow the "Cisco AnyConnect Socket Filter" to filter network content. Click the Allow button.
The AnyConnect VPN client should now work normally.