Perhaps so that if there's a dispute about whether the signatures are genuine, a random sample can be verified to prove it.
Have you ever wondered whether you would have had the courage to join the protesters for freedom, had you lived in 1940s India, 1980s Russia or 1990s South Africa? I like to think I would, but who can say for sure; it must've taken courage far beyond anything being asked of us here and now. But we can at least do what is asked of us.
Because it's a petition, not an online poll. Because petitioner signatures are given to the petitionee, and should represent real people, not faceless internet folk.
If you're not willing to publicly support the cause, you clearly should not sign it. (Although you can choose not to have your name listed on the front-facing site.)
Have you ever wondered whether you would have had the courage to join the protesters for freedom, had you lived in 1940s India, 1980s Russia or 1990s South Africa? I like to think I would, but who can say for sure; it must've taken courage far beyond anything being asked of us here and now. But we can at least do what is asked of us.