Why aren't my triggers firing during an insert by SSIS?

Because the OLE DB Destination task uses a bulk insert, triggers are not fired by default. From BULK INSERT (MSDN):

If FIRE_TRIGGERS is not specified, no insert triggers execute.

One must manually specify FIRE_TRIGGERS as part of the OLE DB component through its Advanced Editor.

Then add "FIRE_TRIGGERS" to the value of FastLoadOptions (note that options are comma-separated):

With that option in place, the triggers should fire during the task's execution.

complementing ladenedge's answer.

Because the OLE DB Destination task uses a bulk insert, triggers are not fired by default

that is true when you have the "fast load" option selected.

If you change it to a regular "table or view" data access mode, your triggers should fire normally because the insert is done row-by-row