What Kirei Kotomine is talking about is allowing Angra Mainyu to be born.
first off, Kirei Kotomine feels nothing out of life and it has been like this since birth. he even tried to start a family to gain a true sense of self but even after fathering Caren Hortensia he remains unchanged. to quote is Personality on the wikia
Unknown to all those around him, Kirei was born defective, possessing a warped mind since birth. He is an evil person, but not a villain. He is deviant, but not inhuman. He is not sane, but neither is he insane. Despite his personal accomplishments, he never felt satisfaction. Despite his best efforts he could not enjoy what other people found happiness in, and instead found himself drawn to negative emotions like the pain of others. He is able to act like a normal person, even bringing pride to his father, but he intrinsically cannot understand others or things they see as "beautiful."
his talks with Gilgamesh in Fate/Zero make him realize he takes "joy" from the misery of others but this questions why someone like him was born.
the scene you link and describe appears to be at the end of the Forth Holy Grail War after the the grail "spilled" dumping the black mud on top him and Gilgamesh.
Kirei: Hahahahaha! This is it?! What the hell does this make me?! What manner of dark evil! What manner of cruelty! This? This is my wish?! This horrible destruction and tragedy?! This is the joy I sought? Could something so twisted… so corrupt… have truly sprung forth from the seed of Father Risei Kotomine?! Hahahahaha… No way! It is impossible! Did my father sire a soulless demon more evil than the Devil himself? Hahahahaha…
Archer: Are you satisfied now? Kirei?
Kirei: No… no, not yet. This isn’t merely enough. Yes, it is true that my life of questions has finally found its answer. However, the problem solving process itself was skipped over and the solution was just simply handed to me. How is it that I’m supposed to accept all this at face value? The formula that yielded this strange, unsettling answer should exist somewhere. As a greater, clearer principle one can understand. No… Not should, it must. I must question… I must search… I must dedicate the remainder of my life... to fully understanding it.
Archer: You know, you never fail to bore me… As you like. Your questions would slay the very gods. I shall watch you seek out your answer.
Kirei1: Hm... Angra Mainyu… I will reach it again. And next time, I will see it to the end. Its birth, the proof of its existence. Hm?
before this Kotomine had been killed but the black mud from the grail which is caused by Angra Mainyu having corrupted the grail (as shown by Kiritsugu's encounter with it in the form or Irisviel and Ilyasviel) was spilled when Kiritsugu ordered Saber to destroy the grail vessel. the black mud washed over Kotomine corpse and Gilgamesh (who is still around because of the Archer Classes Independent Action), the result was Kotomine being resurrected with a new fake black heart and Gilgamesh was given a real body increasing his A Ranked Independent Action to A+ allowing him to not even need a Master (but hangs around Kotomine for entertainment and a source of prana).
in the Heaven's Feel Route of Fate/Stay Night, near to the end he reveals why he's showed up to stop Shirou stopping the Greater Grail which also explains why in the previous routes he tried to summon the grail
When Shirou eventually frees Sakura from Angra Mainyu's grip, Kirei suddenly resurfaces to stop Shirou from preventing the dark entity's birth. He bitterly recounts to Shirou how his lifelong inability to find salvation and happiness has led him to question why someone like him was born. In pursuit of the answer to his lifelong question, he explains that he has dedicated his entire existence to bringing about Angra Mainyu's birth even if it causes his own death and the world's destruction. By witnessing the behavior and stability of an entity born of all the world's evil, he is convinced that he will gain a deeper understanding of not only himself but the world that saw fit to give him life.
Source: Kirei Kotomine - Role - Fate/Stay Night - Heaven's Feel (2nd last paragraph)
so his new raison d'être is to allow Angra Mainyu to be born to then understand why himself was born
1: the site says Kiritsugu here but it's a mistake as Kirei sees Kiritsugu and speaks about him in the next couple of lines.