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.
Well, to get humans to space today you have either the crew dragon on a SpaceX falcon 9, or a Soyouz.
And relations with Russia are a bit tense nowadays, so if you are not Russian, there are currently no other options than SpaceX.
Kinda looks like a monopoly, with another actor starting to compete (and SpaceX is historically very competitive on prices, so without huge government subsidies, I doubt Boeing would consider maintaining their crewed vehicle)