The photo itself was a reenactment of the original event that had taken place the previous day, but during the battle of Berlin there was certainly very fierce fighting in the Reichstag. Although it had not been in use since 1933 when it was burned, entrenched Waffen-SS troops fought hard to keep the building. There was brutal hand-to-hand combat and room-to-room combat, but on May 2nd 1945 the soviets had complete control of the Reichstag, 5 days before the unconditional surrender of Nazi Germany.
Once Soviet troops penetrated into Berlin proper and began overrunning parts of the city Stalin placed great pressure to seize control of the Reichstag and raise a flag over the building by May 1st which was International Workers' Day. The building itself was defended by elements of a severely depleted Waffen SS division. Despite this fact there was no thought given to surrender as the Soviets routinely executed the SS so the fighting was brutal and unrelenting. The attacks on the Reichstag began on April 30 by elements of the 3rd Shock Army under the command of General Colonel V.I. Kuznetsov. The Soviets did manage to place flags multiple times on the Reichstag before the actual photo.
After several false starts and missteps a Soviet soldier, Rakhimzhan Qoshqarbaev managed to place a flag in the crown of the mounted female statue symbolizing "Germania" this was achieved at approximately 22:40 on April 30. However, due to it being night no picture captured the event and the flag was removed by the Germans before morning.
After the complete pacification the official story alleges that Yevgeny Khaldei took a picture of two Soviet soldiers, Meliton Kantaria and Mikhail Yegorov raising the Soviet flag in a quintessential symbol of utter Soviet victory.
Khaldei whose identity was not released until after the fall of the Soviet Union due to the secrecy behind the event disputes this story. He claims that he scaled the Reichstag with the flag in tow and simply selected some soldiers who happened to be passing by to assist him with the photo shoot. Including himself the others were Alexei Kovalyov, Abdulkhakim Ismailov, and Leonid Gorychev. The reasons for this change by the Soviet media were tied to the nationalities of the men involved. Kovalyov was from Kiev, Ismailov from Dagestan, and Gorychev from Minsk. The two men officially given credit were a Russian and a Georgian. Whether Khaldei is telling the truth or remembering it correctly is disputable, but it would hardly have been out of character for the propaganda value of the picture to change the identities of the men involved.