According to Twitch, “Get up to 12 months of access to Nintendo Switch Online at no extra cost. You can claim the 3-month Nintendo Switch Online access offer now, then come back to claim the 9-month individual membership when it unlocks.” Basically this means that upfront you will be able to claim 3 months of free membership, and after 60 days and assuming you’re still subscribed to Twitch Prime, you will be able to claim the remaining 9 months.
For a bit of pricing context, Twitch Prime is part of Amazon Prime that costs $12.99 a month or $119 a year. The Nintendo Switch Online is priced at $19.99 a year. This means that if all you need is Nintendo Switch Online, then it might be cheaper to just pay for it outright rather than purchasing a subscription to Twitch Prime.
However, for those who are already subscribed to Twitch Prime, they will have until the 28th of September to claim the 3 month subscription, and until the 28th of January, 2020 to claim the 9 month subscription.