I believe this was only for customers who opted for the "extended care" package when first purchasing their phones. The envelopes were sent from Asurion and there was zero interaction on my end to facilitate it, they just showed up out of the blue.
I was an early pre-order on the first two phones which did get extended care, and didn't get the extended care package on the third. Only two cases showed up.
I'm pretty sure it's just Asurion doing some math and figuring out that free cases to all their insured users will reduce the number of repair/replace claims by enough of a margin to exceed what they would otherwise spend on those claims.
If you're eligible, I'd say just wait for it to show up.