How does Taric gain bravado?

It actually says it in the description:

After casting an ability, Taric's next two basic attacks within 4 seconds gain 100% total attack speed, each dealing 25 − 93 (based on level) (+ 15% bonus armor) bonus magic damage and reducing the cooldowns of his basic abilities by 1 second.

So after using any of your normal spells (Q,W,E,R) you will trigger the passive which will empower your next two basic attacks. Note that summoner spells and item actives won't trigger Bravado.

With the exception of armor (damage scaling), most stats won't have any direct impact on Bravado but you can affect them indirectly with attack speed and cooldown reduction.

• Cooldown reduction
  This is definitely a recommended stat for taric since it scales both ways with his passive. Casting abilities will trigger your passive which results in more damage and flat cooldown reduction which allows you to cast more abilites (again allowing you to trigger your passive more frequently).
• Attack speed
  Not really a recommended stat for Taric to build in a game where you're in your usual role (support) since it's quite expensive but if you're going for a less serious game he can have some quite surprising dueling potential with it. Attack speed lets you consume your stacks faster which means you will get more value out of them.

In general the purpose of Bravado is to promote a playstyle in which you are weaving in spells between your auto-attacks in order to buff your damage and utility. You can't have more than two stacks at any point in the game but you can get more out of it than the tooltip suggests.

The two "stacks" you see are how many empowered hits ("[...] Taric's next two basic attacks [...]") are available for use. After activating Bravado by casting a spell, you gain two empowered attacks. The counter then decreases after each attack performed.

Bravado is the name of Taric's passive

So to gain Bravado, Taric simply needs to cast a spell. Upon gaining Bravado, his next two basic attacks are empowered. This can be refreshed by casting another spell. Taric cannot gain Bravado stacks (Bravado is not a stacking buff), but it can be refreshed. The icon simply signifies how many remaining empowered auto attacks you have left due to Bravado.

This works the same way as Lee Sin's passive does where his next two auto attacks after casting an ability restore energy and give attack speed.