I think this makes sense for Amazon for two reasons:
1) One of the barriers to entry of live-streaming sites is the expensive hosting costs. With AWS infrastructure, Amazon could potentially have another competitive advantage over other live-streaming sites. They may even provide a better service through higher resolution streaming as they wouldn't be as constrained with bandwidth costs.
2) As Amazon enters the online advertising space to compete with Google Adwords and Adsense, they'll want to own web properties with high impressions for their display ads. User based video creation is great for that but comes with risk for copyright violations. Twitch solves both these issues as it'll give display ads high impressions without much concern about copyright violations as these will mostly be legit user-originated content.
2) As Amazon enters the online advertising space to compete with Google Adwords and Adsense, they'll want to own web properties with high impressions for their display ads. User based video creation is great for that but comes with risk for copyright violations. Twitch solves both these issues as it'll give display ads high impressions without much concern about copyright violations as these will mostly be legit user-originated content.