You don't use the tilt element of tilt-shift lenses to fix perspective, only the shift element.

Tilting tilts (or swings - that's the term for a horizontal tilt) the plane of focus. It's mostly used for increasing apparent depth of field. Imagine you're taking a picture of a football field. You want the entire field to be in focus but you're using a large format camera and you can't stop down too far. Thanks to movements (tilt/shift) you can tilt down your plane of focus, so instead of vertical, it's tilted forward covering the field. You lose focus on the sky, but that doesn't really need to be in focus anyway, right? What's funny is that although you tilt part of the lens, the scene you're framing never actually changes.

All this is covered in this fine wikipedia article on the Scheimpflug principle.

Shifting is pretty simple. Inside your camera, your lens works a bit like a projection lens works on a movie screen. This is why longer focal lengths give you more "zoom" - the same thing happens when you move your projector away from the screen. Your lens projects and image onto your sensor or film - just like you have it projected on a screen in a movie theater. With 99% of lenses, the position of the lens is fixed and the image shown on your sensor is just big enough to cover it... or sometimes, not, that's when vignetting happens. With shift lenses, your circle is much bigger, so when you shift, you really just move the lens around in front of your sensor, as if you were moving a projector.

Shooting large format really helps you understand that a camera isn't a fixed object, it's really two planes that can interact with each other in a huge number of ways.