This allows you to delete old messages from the app. If you have an online/Pulse SMS+ account, it also deletes them from the web, since the web version of Pulse SMS is a direct mirror of what is on the phone version. This feature comes in two different flavors:
Globally:
In Settings > Advanced Features, you can select a time interval and Pulse SMS will delete messages older than that time. This service runs once per day and will delete the messages on the phone, as well as your other devices. This cannot be undone.
Within a Conversation:
From within the Conversation settings of any individual conversation, you can select this option to delete older messages within that single conversation. This is useful if you have a conversation that is getting very large and you want to speed up loading it. This option also allows you to select a time interval. Once you select the interval, it will run immediately and delete the old messages on the phone and across all your other devices. There is no way to undo this. Unlike the global option, it doesn't repeat on the next day, i.e. you will need to access it once more in Conversation settings should you want to use it again.