@Gargron I usually tell them about the #Fediverse first. And then I saw #Mastodon is the software I'm using.
It might seem silly, but one of the main arguments for me compared with #Twitter is that Twitter decides which tweets you see and can thus influence your and other peoples opinions. While Mastodon shows objective timelines.
Another one is that your account can't be closed by whomever has power over Twitter.
(instance image by мøтħer ¢røω)