-Ezreal is a champion highly based on trinity force, which gives him burst. Both AD and AP items make the burst more dangerous by simply making it bigger.
-Ezreal has really nice dmg output with his skillshots skills. All of them (Q,W,R) have scaling ratios with both AP and AD.
-Besides the fact that you get more dmg with it, fully stacked Mejai's soulstealer gives you extra cooldown reduction which is an amazing thing for Ezreal. Bigger cooldown reduction = more skills = bigger dps = win
-Ezreal already has an amazing escape ability (E) which allows him to quickly escape any dangerous situation which makes him really hard to catch. Now when you add some movement speed to that + mejai's cooldown reduction + occult's movement speed + extra movement speed with every hit spell or autoattack it makes him practically uncatchable.

This build is not preffered for a normal game though. Achieving 20 stacks on Mejai's and Occult is really hard to get, and it is really easy to lose them. If you really want to try that build i recommend playing with friends who will help you achieving them. A kill - 2 stacks. An assist = 1 stack. A death = loss of 1/3 collected stacks. It is really hard to make this build effective in the beggining, but if you are bored with typical Ezreal, this build is really fun to play and i highly recommend it. Just for fun. :)