I'll explain as if your first language is not English.

Dry means something that has no moisture in it, like sand can be dry or wet.

Obviously something dry contains no moisture or liquids.

A Martini is a cocktail, mixed from mostly gin and vermouth. In this context, dry means "little vermouth", so the taste of the gin is most prevalent.

So the joke is between the two different meanings of the word, as obviously a dry cocktail makes no sense if you take the first meaning of the word.