What is the wisdom behind the legislation of the prostration of recitation? What is its evidence and what is the manner of performing it?  

Allah Almighty has legislated the prostration of recitation (sujud al-tilawah) to manifest complete submission to Him. This occurs when a Muslim recites or listens to a verse whose meaning calls for prostration to Allah Almighty. It is more strongly emphasized for the reciter of the Quran than for the listener.

The basis for its legislation is found in the words of Allah Almighty Who says, “Then what is [the matter] with them that they do not believe? And when the Quran is recited to them, they do not prostrate [to Allah]” (Quran, 84:20–21|) and “Indeed, those who were given knowledge before it—when it is recited to them—they fall upon their faces in prostration”
(Quran, 17:107).

Abu Huraira (may Allah be pleased with him) narrated that the Messenger of Allah (peace and blessings be upon him) said, “When the son of Adam recites a verse of prostration and prostrates, Satan withdraws and weeps saying, ‘Woe to me! The son of Adam was commanded to prostrate and he prostrated, so he will go to paradise. I was commanded to prostrate but I refused, so I will go to hellfire”
(recorded by Muslim in his Sahih).

The proper way to perform the prostration of recitation
The reciter or listener should say ‘Allahu Akbar’ and perform a prostration — whether from a standing or sitting position — without bowing. Upon rising, they should say ‘Allahu Akbar’ again, without reciting the tashahhud (the testimony recited during prayer) or saying taslim (the concluding salutation that ends the prayer).

And Allah the Almighty knows best.