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It's basically for older players (i.e. Fowler, Hasselbaink, etc) who want to be paid a large amount, but might not be able to play every week, so they pay them more appearance bonuses and less weekly. Dunno why middle age players get them though
I think by offering them an appearance fee, they would negotiate a lower base wage in exchange.
Also Liverpool are more likely to rotate. And they have brought in a lot of new players each season so it is less predictable what their first team is going to be from season to season.
So when Riise last signed his contract. He may not have been an automatic starter at that time.