The Like List is a list of cache logs that have been marked as "liked" by cachers.
You mark a log as liked by clicking on the green thumbs up icon when you view a log.
You can remove a like on a log by clicking on remove on your Like List.
You can see your own like list on your hotlist http://geocaching.com.au/my/hotlist/likelist/
This list also contains the logs that have been liked by others.