Joe Rosenthal on taking the photo June 25, 1975, the interviewer, Benis M. Frank, speaks in brackets (Source and additional resources at bottom):

‎ ‎ ‎ "[Can you describe the progression of events that
brought you to the top of Mount Suribachi and in position to capture the famous
flag-raising image?]

‎ ‎ ‎ My stumbling onto that picture was, in all respects, accidental. I had been out to the command ship the evening before and was headed back toward shore. And as it got close, a boatswain on an LCT [landing craft, tank] told us, Bill [Hipple, correspondent for Newsweek] and me, that they heard over the radio that there was a flag—American flag—going up to the top of the mountain [Mount Suribachi]. I couldn’t see it out there, but someone said he thought he saw a spot up there. Well, “I’d better go take a look.” Now, there were other correspondent-photographers along the way, but I didn’t know where they were.

‎ ‎ ‎ B‎ut I went up, and I was late [by] at least an hour—an hour and a half—as they pinned it down later on. I found that the official time for that first flag going up there was about 1035 in the morning. And near as I can pin down, the time . . . this is by consulting with a priest who said mass up there, and various other things, which I did objectively I mean, I went backward. I found that my picture was taken closer to noon. Anyway, when I heard that, I thought I’d better go up and take a look. Now, I heard that it was a patrol of four guys that had gone up there. That’s the way I heard it. By the time that I got there, there was, more or less, a stream of guys going up. It’s hard climbing. Every once in a while, we’d have to duck because there’d be [someone would shout] “fire in the hole” and throwing grenades in the caves. But as I approached the brow of the hill of Suribachi, I saw guys about a 100 yards away from the flagpole that was already up. And it was a fairly small flag on a long pole, and it was [waving] in the breeze. There again I felt one of those little clutches at the heart—this was our flag. I’m still touched by recollection. Stopped there for a moment.

‎ ‎ ‎ And as I came closer— There were several Marines [who] were kneeled on the ground, and one of them had a folded flag in the traditional triangular folded [shape] under his arm. And they had a pole out there, and they were fiddling around with some kind of wire or rope—whatever it was—and I said, “What’s doing, fellows?” They said, “Well, we’ve got to take down the other flag and keep that as a souvenir, and put up this larger flag so they can be seen better around the whole of the island.”

‎ ‎ ‎ That gave me time to walk around, and I made a survey. Now, I had no idea of any great importance to be attached to those pictures. It was an incident during the battle. I wasn’t thinking, of course, in any terms that it would be a lasting picture, an inspiring thing, or any of that. It was one of a number of pictures that I was taking during the course of the battle. Some of them were tense situations; some of them were, more or less, pictorial in nature and so on. First of all, I tried to find the four guys [who] had been mentioned. I had some names. I looked around; I couldn’t find them and I got no help from the lieutenant either. You know, he had things to do; he was busy. There was still shooting into caves. There was still a lot of potential opposition all around the place. That was [Harold G.] Schrier. . . . I don’t mean it as critically as it might sound, because I do appreciate that he had things to do.

‎ ‎ ‎ So I attempted to get those fellows; but it seemed to be too difficult to do and I just, again, I passed over it. I’m not thinking of something of any great importance anyway. So as I walked over and took a position where I could estimate that I would get the whole throw of this pole upward with that larger flag. I looked around and I spotted a couple of sandbags that raised me a foot or so, and I’m already pretty close to the ground. Well, it was enough, and I simply waited for a moment or two.

‎ ‎ ‎ [Now you are in position to take a photo. What happened next?]

‎ ‎ ‎ By that time, Bill Genaust, the Marine photographer, came across in front of me and over to my right, just an arm’s length. He said: “I’m not in your way, am I, Joe?” And I said, “Oh, no.” I turned from him, and [looking] out of the corner of my eye, I said, “Hey, Bill, there it goes!” By being polite to each other, we both damn near missed the scene. I swung my camera around and held it until I could guess that this was the peak of the action, and I shot. Of course, I couldn’t say positively I had the picture something like shooting a football play; you don’t brag about it until it’s developed.

‎ ‎ ‎ Notice the historic moving sequence that he [Genaust] shot on this in color film— You’ll note one thing: that that action of that flag raising starts right now. There’s no preliminary footage. That’s the reason that we both almost got caught short by this Alphonse-Gaston act. Bill was that kind of a guy. He was a high-grade guy. Bill was killed a few days later, a week or so later. He got too close to a cave entrance where Japanese [soldiers] had holed up, and [they] got him caught right in a cross fire. Now they shot him, and they pulled him inside the cave. I went down to inquire about him a couple of years after, and also 10 years after, and I could not find the grave registration for him. I think it was one of the caves that they simply had to seal up...."

(See more in source, further questions asked are "When did you finally see the photograph? What was your reaction?", "Were you involved in any of the public relations activities featuring the flag-raising image?", and "How did the photograph affect you personally? What is its significance?")
‎
Source:
‎
Transcribed interview

"U.S. MARINE CORPS
HISTORY DIVISION
ORAL HISTORY
INTERVIEW"

In

INVESTIGATING IWO
The Flag Raisings in Myth, Memory, & Esprit de Corps
‎
Contributing editor
Breanne Robertson, PhD
‎
https://www.usmcu.edu/portals/218/investigating%20iwo_web2.pdf Transcribed Interview starts on page 315 (344 in pdf)

Additional resources on the photo:

U.S. flag raised on Iwo Jima (The History Channel) - https://www.history.com/this-day-in-history/february-23/u-s-flag-raised-on-iwo-jima

USA: IWO JIMA PHOTOGRAPHER - JOE ROSENTHAL (AP Interview) - https://www.youtube.com/watch?v=37Br0CBBxjM

Joe Rosenthal and the flag-raising on Iwo Jima (Pulitzer) - https://www.pulitzer.org/article/joe-rosenthal-and-flag-raising-iwo-jima

Oral History Interview with Joe Rosenthal (Texas Wesleyan University) - (https://texashistory.unt.edu/ark:/67531/metapth1607506/m1/

Raising the Flag on Iwo Jima: Behind the Photo (History Channel) - https://www.history.com/articles/iwo-jima-flag-raising-marines-photo

Raising the Flag on Iwo Jima (Wikipedia Page) - https://en.wikipedia.org/wiki/Raising_the_Flag_on_Iwo_Jima