The activity log in WHMCS records events that occur to allow for tracking who and what initiated various actions and to allow for troubleshooting any errors or problems that occur.
For example, if automatic setup ever fails for any reason, the activity log will hold the error message response that came back from the module to allow you to find out why. Similarly, if you have any problems with the cron, the activity log will show you what tasks the cron is performing and allow you to see at exactly which point or item it is failing.
Reviewing Activity Logs
There are 2 types of log entries, client logs and system logs. Client logs are events and actions relating to a specific client and can be viewed from the Log tab within a Clients Profile. Alternatively the full system log which includes both client and system log entries can be accessed via Utilities > Activity Log
Each action/event records the date/time the action occured, a description indicating what happened, the currently logged in admin user who invoked the action (System shows for non admin initiated actions) and the IP Address of the user who caused it (ie. the admins IP, the clients IP or the system IP).
In both locations, you can filter the log entries for entries on a specific date, performed by a specific username/IP, and/or containing certain keywords you specify. This allows you to narrow the log down quickly and easily for the specific action or event you are looking for.
The system log entries are automatically purged and kept to the limit set in Setup > General Settings > Limit Activity Log, while client log entries will be kept indefinately so the logs can easily reach many thousands. This shouldn't affect the performance of your system in any way but if you do want to trim client history logs then there's an option to delete all entries before a certain date to prune the logs in Utilities > System Cleanup