But how does the Commander know that "the dude" with the hoodie was Palpatine or did he already know Darth Sidious?

He didn't need to...

Obeying Order 66 was pre-progammed into all clones of which Cody was one.

Control chips, also known as inhibitor chips, were a type of organic bio-chip capable of dictating or responding to the thoughts of its host. Kaminoan cloners implanted them within each clone trooper in the Grand Army of the Republic. It was implanted into the brain of every trooper at the third stage of their embryonic development. When they were activated the troopers responded to orders without question, allowing the Supreme Chancellor to initiate Order 66.

^Wookiepedia

So, the Chancellor activated the chips and Cody had no choice but to obey.

As a result of the commander's programming, Cody obeyed the chancellor without hesitation .

^Wookiepedia

As detailed in Valorum's answer to a similar question on SF&F

The film's official novelisation indicates that the frequency was reserved for Palpatine.

“Yes, sir.” A silent buzzing vibration came from a compartment> concealed within his armor. Cody frowned. “Go on ahead, General. We’ll be right behind you.”

That concealed compartment held a secure comlink, which was frequency-locked to a channel reserved for the commander in chief.

^{Star Wars Episode III: Revenge of the Sith}