Best Answer - Chosen by Voting
'To log' is short for 'to write down in a log book'. A log book is a record of activities.
'Hours' here would be the length of time spend doing some activity.
As an example - long distance truck drivers are required to keep a record of the hours they spend driving, so that they can prove that they had had enough rest, and weren't getting too tired.
They would refer to 'logging their hours' when they fill in the records in the log book.
