The new book "No Easy Day" by former U.S. Navy SEAL Matt Bissonnette, who wrote under the name Mark Owen, gained widespread attention because of his firsthand account of how he and other members of SEAL Team Six killed Osama bin Laden.

On Sunday night, Bissonnette shared more of the intimate details of the mission in an interview with CBS's "60 Minutes."

Bissonnette wore heavy makeup and his voice was disguised as he described what he said was not just a "kill-only" mission, but a chance to capture the mastermind of the September 11 attacks alive, if possible.

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"We weren't sent in to murder him. This was, 'Hey, kill or capture,'" he told interviewer Scott Pelley. Bissonnette said that in the weeks leading up to the mission, the SEALs trained on a full-size model of the compound in Abbottabad, Pakistan, where they would eventually kill bin Laden. It was rare, Bissonnette said, to get 100 chances to train on a mock-up like that for three weeks.

Bissonnette said that while it was the most important mission he would ever be a part of, much of what the team members did was routine, until the moment they could finally exhale, knowing they had killed their biggest target.

Below are some of the most interesting exchanges between Bissonnette and Pelley, according to CBS transcripts, about the preparation for the mission, the raid itself and his reaction to it all when it was finally over.

On how they cleared the house as they hunted for bin Laden after taking early fire:

Matt Bissonnette: Guys start making their way up the stairs. And it's quiet. It's pitch black in the house. No lights. All night vision. Get to the second floor. Intel had said, "Hey, we think that Khalid, his son, lives on the second floor."

Scott Pelley: This is Osama bin Laden's son?

Bissonnette: Yeah. The guy in front of me who is point man, he sees the head pop out and disappear really quick around the corner. He's like, "OK, you know, what - who is it? What do you think?" "Yeah, I don't know." He literally whispers, not amped up, not yelling, not anything. He whispers, "Hey, Khalid. Khalid." He whispers Khalid's name. Doesn't know if it's Khalid or not. Khalid literally looks back around the edge of the hall. And he shoots him. What was Khalid thinking at that time? Look around the corner. Curiosity killed the cat. I guess Khalid too.

Pelley: Somebody started shooting at you from inside the house? And the bullets were coming through the door?

Bissonnette: Yep. Immediately, my buddy who was standing up started returning fire. I could - yeah, I kind of rolled away from the door, blindly returned fire back through. You couldn't see what was on the other side. And then it went quiet. Thankfully, the SEAL that was there with me, that initially returned fire with me spoke Arabic. So he immediately started calling out to the people inside. Started hearing the metal latch on the inside of the door. Are they gonna come out with a suicide vest? Are they gonna throw a hand grenade out? Are they gonna, you know, spray their AK? Door opens up, a female holding a kid, couple kids right behind her.

Pelley: You got your finger on your trigger and you're looking at a woman with her children?

Bissonnette: Yeah, yeah. Split second. I mean, we had just received fire. My buddy's speaking Arabic. And he's asking her, you know, "Hey, where's your husband? What's going on?" She - and - and she replies back to him, "He's dead. You shot him."

On how they killed bin Laden, but weren't sure it was him:

Pelley: Khalid is dead on this landing. The point man is stepping past Khalid. And now, you're No. 2 in the stack. You're right behind the point man?

Bissonnette: Yep. I'm kinda trying to look around him. Hear him take a couple shots. Kind of see a head - somebody disappear back into the room.

Pelley: The point man had seen someone stick his head out a door and shot him just the way he'd shot Khalid.

Bissonnette: Yup.

Pelley: What did you do then?

Bissonnette: Inside the room, I could see a body laying on the ground. Over him was two females, real close to the door. They looked up and saw the point man. He steps into the room, literally rushes the two women, grabs one under each arm, and pushes them back against the far wall. So if they did have a suicide vest on, and they did blow themselves up, that they wouldn't - that wouldn't affect the rest of the guys.

Pelley: But it would have killed him?

Bissonnette: Yeah.

Pelley: You stepped into the room and saw the man lying on the floor? What did you do?

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Bissonnette: Myself and the next assaulter in, we both engaged him several more times and then rolled off and then continued clearing the room.

Pelley: When you say you engaged him, what do you mean?

Bissonnette: Fired.

Pelley: You shot him?

Bissonnette: Yeah.

Pelley: He's still moving?

Bissonnette: A little bit. But you couldn't see his arms. Couldn't see his hands. So, he could've had something. Could've had a hand grenade or something underneath his chest.

Pelley: So, after Osama bin Laden is wounded, he's still moving. You shot him twice?

Bissonnette: A handful of times.

Pelley: A handful of times, and the SEAL in the stack behind you also shot Osama bin Laden. And at that point, his body was still?

Bissonnette:: Yes.

Pelley: Did you recognize him?

Bissonnette: Nope. You know, everybody thinks it was, like, you know it's him. No. To us, at that time, it could have been anybody. Maybe this was another brother. Maybe this is a bodyguard. Maybe, it doesn't matter. The point is to just continue clearing.

On how they initially confirmed they had just killed bin Laden:

Bissonnette: So he moved out to where the women and kids were, grabs one of the younger kids. Says, "Hey, who is that inside?" She says, "Osama." "Osama who?" "Osama bin Laden."

Pelley: The child?

Bissonnette: The child.

Pelley: Identified him?

Bissonnette: Yep. Grabbed one of the females, again asked her, "Hey, who is that?" She said, "Osama bin Laden."

Pelley: Does a cheer go up among the SEALs? You start shaking hands? Patting each other on the back?

Bissonnette: Not - nothing. It's all business. We're on the clock here. So, we call up the commanding officer. He comes upstairs. Looks at the body. We give him what we have so far of - hey, here's what he looks like, take a look. He's tall. Woman and kid confirm it. He took one look. He said, "OK, I think that's him."

On what Bissonnette made of bin Laden not making a final stand:

Bissonnette: And some people would argue that, you know, why did that point man take those shots? Well, immediately, the first door we went to, my team was engaged by enemy fire through the door. So automatically, we know we're going into an enemy compound, shots being fired back at us immediately. AK found next to Khalid on the stairs. All those boxes have been checked that if a guy sticks his head around the corner, he very easily could have a gun. You don't wait to get that AK or the grenade thrown down the hall or the suicide vest. So in the split second, that's when he engaged.

Pelley: He did have a gun. But he didn't use it. And I wonder what you make of that?

Bissonnette: I think in the end, he taught a lot of people to do - you know, martyr themselves and he masterminded the 9/11 attacks. But in the end, he wasn't even willing to roger up himself with a gun and put up a fight. So I think that speaks for itself.

On how he saw the news announced to the world - and then modestly celebrated

Bissonnette: Yeah, we watched it live. It was - they had some TVs set up in the hangar that we were at. And, literally still in your camouflage uniform, our gear kind of set to the side, and we heard it was coming on. We went and gathered around and watched the address.

Pelley: What did you think?

Bissonnette: Now the world knows that we've got him.

Pelley: When you landed back in the United States, what did you think of all the media coverage?

Bissonnette: It was all surreal because, you know, this had all been so hush-hush leading up to it. We went and did it. And now it was the biggest news story ever. We got on a bus. They drove us back to work. I didn't even go in. They told us we had a couple days off. And I grabbed my keys, went and got in my truck and, you know, I put it in the book. But, you know, I hit Taco Bell on the way home, hit the drive-through, a couple tacos. And, you know, ate it in my car right there and then drove home.

Pelley: You were part of the team that killed Osama bin Laden and the first thing you do when you get back to the United States is go to Taco Bell?

Bissonnette: Two tacos and a bean burrito. It's routine.

Read more about the book, the debate over it being published and the bin Laden raid:

SEAL's Osama bin Laden book sheds light on Ambien use in military

Pentagon: SEAL's book has classified info, didn't follow protocol

Former SEAL worried about new book

Excerpts from SEAL's book about Osama bin Laden killing