According to Scott Mingus Jr., writing in "Teddy Roosevelt and the Rough Riders at Kettle Hill" at MilitaryHistory.com: "Around noon [on July 1, 1898], the First Infantry Division began their assault on San Juan Hill with the support of General Joseph Wheeler's Cavalry. Seeing the attack on adjacent San Juan Hill starting, Roosevelt still in position in reserve below Kettle Hill became more and more impatient. Finally he was given the command to start the Rough Riders' assault up Kettle Hill. Roosevelt, feeling very excited, began to ride up and down the line urging the men forward. Seeing Roosevelt personally urging them forward, momentum quickly spread and finally the entire regiment was on the move forward, passing the forward lines of other American units...
"As the American cavalry and infantry reached the lower slopes of the hill, Spanish fire became more deadly and more accurate. But... the Rough Riders and their supports continued to steadily climb the hill. As the Americans reached the second fenceline, it became apparent to the Spanish defenders that to avoid hand-to-hand combat, they must withdraw. ...then they began to withdraw.'
"Reaching the top of the hill, the Rough Riders established some initial defensive measures to secure the area as the Spanish on the higher San Juan Hill began to fire upon them. Seeing that the attack on the adjacent height was not going too well as heavy fire and other obstacles were stopping the forward progress, Roosevelt himself decided to lead a charge, a decision reinforced when three Gatling guns got to the top of Kettle Hill to support an attack.
"Shouting for his men to follow, he jumped over a barbed wire fence and ran down the slope. Roosevelt ran about one hundred yards when he turned around to notice that only five of his troopers had followed him down the slope into the swale between Kettle and San Juan Hills. Roosevelt turned around and returned to the crest of Kettle Hill where his troopers claimed they did not hear his order to charge. Forming them quickly into an assault line, Roosevelt again ordered the charge.
"This time, the entire regiment with support from other nearby forces went forward towards San Juan Hill.... By two thirty in the afternoon, the entire heights were possession of the American troops. With the victory at San Juan Heights, the Americans were able to move into the city of Santiago and establish a good offensive position to fire on the Spanish fleet in the harbor. With the subsequent destruction of the Spanish fleet from an attack from Rear Admiral William T. Sampson, the Americans were able to win the battle, and force and early end to the war.
"Theodore Roosevelt's personal actions at the battle of San Juan Heights would win him the nation's highest military honor, the Congressional Medal of Honor.... His enthusiasm, confidence, bravery, and high regard for his troopers would him enduring fame with his Rough Riders, who associated victory solely with Roosevelt, not with their actual commander Leonard Wood, who has all but been forgotten by history."
Curiously, Roosevelt didn't receive the Medal of Honor in his lifetime; he wasn't nearly as popular with the establishment in Washington as with the public after his deeds on the San Juan Heights. The medal was awarded to him posthumously by President Clinton toward the end of his administration in January 2001. In receiving it, TR became the only president ever to win a Congressional Medal of Honor.