To consume a beverage of some sort.
Technically it should only be used in reference to drinking beer because of the history of the phrase:
Back in the middle ages when beer was sold in ceramic cups they would be fitted with a small whistle which could be blown upon finishing ones beverage in order to alert the bar staff that you require another beer. Therefore to get a beer would literally be "wetting ones whistle"
Martin - oh look, I finished my lovely pint of ale. Best wet my whistle with another Broadside or London Pride.
Alexi - Get me some chillie nuts or prawn cocktail crisps while your there, you slag.
To consume a beverage of some sort to cleanse one's pallet.
is a common choice or crunk juice
Brad: Oh boy, am I thirsty! I need something to Wet my whistle!
Carson: Want some water?
Brad: I sure do!
Basically, it means to do something you enjoy. Technically, it means you do something to moisturize your mouth, like drink something.
I'm going to drink a beer to wet my whistle.
I like that car, it really wets my whistle.