The only information we have about this is from the Histories and Lore videos as it is never expanded on in the show itself... that I can find. The full quote from the video you mention offers no explanation.
I was never supposed to be on the Kingsguard. Oh, as a boy, I dreamed of the white cloak like all boys. But I was heir to Tywin Lannister, Hand of the King. If he forbade the tides, the waves would cease. Then I was 15, and my father was congratulating me on my new knighthood in the Red Keep. I wonder if that was the last time he was proud of me.
That night, there was a knock on my door, and I'd opened it to find my sister Cersei disguised as a simple serving girl. I hadn't seen her since my father took her to court when she was 12. She had grown up. As had I. She told me that my father planned to marry me off to Lysa Tully. But she could arrange for the King to raise me to the Kingsguard so I could stay in the city with her. All I had to do was agree. I made the obvious objections. Our family, our father, Casterly Rock. Until she asked, "Is it a rock you want, or me?" Come morning, she had my consent. I would join the Kingsguard for her. I would forswear my lands and title for her. I would forsake our family for her.
Soon, a royal raven commanded my father to present me to the King during the Great Tourney at Harrenhal to say my vows. My father erupted in fury. He could not object openly, but he resigned the Handship and returned to Casterly Rock, taking Cersei with him. Instead of being together, Cersei and I exchanged places.
Game of Thrones, Histories and Lore, "Robert's Rebellion" (Jaime Lannister's perspective)
Even looking into the relevant passages in the books yields no further information, here is Jaime's POV thought process on what happened.
Jaime, meantime, had spent four years as squire to Ser Sumner Crake-hall and earned his spurs against the Kingswood Brotherhood. But when he made a brief call at King's Landing on his way back to Casterly Rock, chiefly to see his sister, Cersei took him aside and whispered that Lord Tywin meant to marry him to Lysa Tully, had gone so far as to invite Lord Hoster to the city to discuss dower. But if Jaime took the white, he could be near her always. Old Ser Harlan Grandison had died in his sleep, as was only appropriate for one whose sigil was a sleeping lion. Aerys would want a young man to take his place, so why not a roaring lion in place of a sleepy one?
"Father will never consent," Jaime objected.
"The king won't ask him. And once it's done, Father can't object, not openly. Aerys had Ser Ilyn Payne's tongue torn out just for boasting that it was the Hand who truly ruled the Seven Kingdoms. The captain of the Hand's guard, and yet Father dared not try and stop it! He won't stop this, either."
"But," Jaime said, "there's Casterly Rock..."
"Is it a rock you want? Or me?"
He remembered that night as if it were yesterday. They spent it in an old inn on Eel Alley, well away from watchful eyes. Cersei had come to him dressed as a simple serving wench, which somehow excited him all the more. Jaime had never seen her more passionate. Every time he went to sleep, she woke him again. By morning Casterly Rock seemed a small price to pay to be near her always. He gave his consent, and Cersei promised to do the rest.
A moon's turn later, a royal raven arrived at Casterly Rock to inform him that he had been chosen for the Kingsguard. He was commanded to present himself to the king during the great tourney at Harrenhal to say his vows and don his cloak.
Jaime's investiture freed him from Lysa Tully. Elsewise, nothing went as planned. His father had never been more furious. He could not object openly—Cersei had judged that correctly—but he resigned the Handship on some thin pretext and returned to Casterly Rock, taking his daughter with him. Instead of being together, Cersei and Jaime just changed places, and he found himself alone at court, guarding a mad king while four lesser men took their turns dancing on knives in his father's ill-fitting shoes.
A Song of Ice and Fire, A Storm of Swords, Jaime II
As nothing is known canonically that I'm aware of it is time for some speculation of what Cersei actually did.
Had an open conversation with the King - seems an unlikely route to take but it could have happened.
Spoke about Jaime and the Kingsguard near Aerys - this seems quite likely, she could have just planted the idea into Aerys' mind by talking about it near him.
Performed sexual acts for Aerys - certainly in keeping with Cersei and her going to great lengths to get what she wants but I'm not sure it would have happened.
Nothing - She could have overheard that Jaime was already in line to join the white cloaks or just assumed he would be.
Ask someone else to talk to the King - She could have approached someone else in court, perhaps on the Small council, and asked them to consider Jaime. This wouldn't have been too hard to do without Tywin finding out as it is very likely he had a lot of enemies at court and so they probably wouldn't have passed the information on.
Now to answer your other questions. Why are they so mad because of this? Well the above quotes make it abundantly clear why:
Cersei and Jaime rejected to it because they were now forced apart again and now on the context of not being able to be back together again.
Tywin rejected because Aerys now removed his heir to Casterly Rock and replaced it with Tyrion - a dwarf that Tywin really would rather see anywhere but that position.
Was Rhaegar already married at this point?
Yes, Rhaegar was married to Elia in 280 AC and Jaime was appointed to the Kingsguard in 281 AC at the Tourney at Harrenhal. It is not exactly clear where in the timeline Cersei came to Jaime with the proposition but I believe he was already married at that point.
Early in the year 279 AC, Rhaegar Targaryen, Prince of Dragonstone, was formally betrothed to Princess Elia Martell, the delicate young sister of Doran Martell, Prince of Dorne. They were wed the following year, in a lavish ceremony at the Great Sept of Baelor in King's Landing, but Aerys II did not attend.
A Song of Ice and Fire, The World of Ice and Fire, The Targaryen Kings: Aerys II
It was the year of false spring, and he was eighteen again, down from the Eyrie to the tourney at Harrenhal.
A Song of Ice and Fire, A Game of Thrones, Eddard XV
In the annals of Westeros, 281 AC is known as the Year of the False Spring.
A Song of Ice and Fire, The World of Ice and Fire, The Fall of the Dragons: The Year of the False Spring