Oh yeah, it's true. The assassin probably would have been successful too, except the bullet went through Teddy's coat pocket, and fortunately for him, inside his pocket was his steel glasses case and his 50 page speech, which slowed the bullet enough that it didn't enter his lungs or heart. I learned about this story soon after I got my first pair of glasses as child, and I thought it was really cool that he was saved by his glasses case, so that's why I remember it.
The only thing that's not entirely accurate from the video is that the speech was
originally supposed to last around 90 minutes, but Teddy explained to the crowd that he had been shot, so he probably wasn't going to get through the whole thing, but he was going to try his best. And he spoke for around 50 minutes before closing out the speech and then receiving medical attention. He reasoned, correctly, that if he wasn't coughing up blood, then the bullet hadn't hit his lungs, and so he figured that as long as he didn't bleed too much, he was okay to deliver the speech.