This is typically a problem with permissions.
Use the Finder to browse to Applications. Select the ZBrush folder (not something inside the folder). Press Cmd+I to get info on the folder.
At the bottom right corner of the Info window there’s a padlock icon. (If your permissions section is collapsed, you’ll need to expand it to see the padlock.) Click that and enter your admin login info.
Now select Everyone from the users list and change that permission to “Read & Write”.
Next, click the gear beneath the list and choose “Apply to enclosed items”.
Once done, lock permissions again and close the window.
If it already says Read & Write for Everyone, change it so something else, apply to enclosed items, change it back to Read & Write, then apply to enclosed items again.
When you start ZBrush again you will most likely need to use Web Activation again. After filling out the form you’ll get a message that “ZBrush has already been activated for this machine”. Click the button to request a copy of the activation key, which will generate a new activation code without deducting another activation from your serial number.
Hopefully that will resolve the issue. If it doesn’t you’re going to need to submit a Support ticket. Be sure to include your serial number and mention that you followed the steps in this thread.