Takedowns on enemies grant permanent Attack Speed.
Azir is a champion who rapidly casts abilities thanks to his auto-attack nature with [W]. Because of this he is able to repeatedly cast Comets with relatively high hit-rate thanks to the slow on his [Q]
Azir is a champion who typically doesn't opt for a mana item, rushing Nashor's tooth and Magic Penetration. Manaflow allows you to supplement that loss of mana sustain and continue pumping out damage in the early to imd game.
Transcendence allows you to opt for a more AP/Magic Pen focused build in the mid game for additional damage output in teamfights.
Scorch provides fantastic early game harass potential so that you're able to keep your enemy laner in kill threat at all times with your jungler.
Triumph's mid-fight heal is fantastic for sustaining long teamfights where you're able to pick up kills/assists and continue pumping out damage.
Azir massively benefits from the extra Attack Speed from Alacrity, empowering your ability to get more soldier attacks off during your [Q] slow.