Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Alright, so unlike most Ezreal builds that focus around raw AD this build works straight around your mystic shot.
Rushing a sheen on Ezreal is almost essential for your damage output. The Sheen procs on your mystic shot cast and the hit effect. This allows you to basically double the damage of your mystic shot.
Next the grabbing of the brutalizer for the attack damage and cool down redction. With this you get that tiny CDR for your mystic shot and arcane shift.
Now rushing your trinity force gives you the amount of damage you need. You won't have to crutch red buff for a slow effect. It's in-general the best item to get on Ezreal.
The next item after your trinity force depends completely on the enemy team. If they're all mages, grab a hexdrinker, if not go for the Infinity Edge.
Your last item should be a ZR to give you some usage on the arcane shift and trueshot barage.