By linking your Twitch Prime account with your League of Legends account, you will receive a free Legendary Shard Skin, but the process can be a little complex.
The first step is obvious, as you will of course need to have or create a Twitch Prime account. This is effectively an Amazon Prime account linked to your Twitch - you can do this here.
Once you have your Twitch Prime account, you will then need to link this to you League of Legends account - you can do this here.
Once everything is linked up, you will be able to claim a free Summoner's crown capsule, containing a random legendary-tier skin shard.
Even if you are not that bothered about this first free reward, it is still worth linking up your accounts, as Twitch says this is only "the first in a series of free in-game loot in League of Legends for Twitch Prime members."
It is being done in promotion of the festivities at TwitchCon 2018, where LoL fans can attend the LAN Arena to play and stream and meet their favorite LoL streamers.
— League of Legends (@LeagueOfLegends) October 26, 2018
On October 28, a special LoL event will be taking place at TwitchCon, called the "Rumble on the Rift", a tournament featuring "the game’s biggest streamers in one place."
Once you have claimed the offer on the website for you free Twitch Prime loot, it will be available the next time you log in, provided you are at least level 3.
HOW TO CLAIM YOUR LEAGUE LOOT
1) Create a Twitch Prime Account
Start your free trial here and become a Twitch Prime member. If you’re already a Twitch Prime member, move onto step 2.
2) Link your League of Legends account and Twitch Prime Accounts
Link your accounts here.
3) Claim the offer
Sign in to your Twitch Prime account here and click "Claim Offer" below the League of Legends offer to claim your Summoner’s Crown Capsule. Your loot will be available the next time you log into your League of Legends account.
Source: League of Legends