How does FILLFACTOR reduce page splits? The default FILLFACTOR is 100 and I have been told at times to reduce it to 80 or lower so that page splits will be reduced due to the empty space on each page.

I don't understand this--if a new index is created on a table with a default FILLFACTOR of 80, 20% space of each page is left empty.  So assume a row needs to use 10% of a page, so when it's full it can hold 10 records.  With a FILLFACTOR of 80, now it can only hold 8 records.  So when you add the 9th record, will there be a  page split?  With 100% FILLFACTOR, a page split will only happen on the 11th record, so it seems that a lower fill factor makes it worse.
The fill factor does not apply to new rows being added to a page.  The fill factor only applies to how full a page is filled when the index is created, rebuilt, or reorganized.

Accordingly, if the index is rebuilt or reorganized on a scheduled basis, your index will be able to accommodate new rows with fewer page splits if the fill factor is set appropriately.