The following explanation is conceptually correct but there are many variations. This is the general idea.

High Definition in our world is simply a relative term to mean an image that has higher resolution and is consequently sharper than a Standard Definition image. Here is why the image is sharper.

Televisions do not have a solid block of information from the top of the screen to the bottom. The picture is made up of little bits of information called pixels placed along lines going across the TV screen. If you look closely at you TV you will see lines

As a comparison, film is not made up of solid information either. A film frame is made up of zillions of little pieces of picture infomation called grains. The grains on film are the equivalent of pixels on the TV image, except there are many more grains on a 35mm frame than there are pixels on a frame of video, even on HD.

On Std Def, there are 525 lines of information going from the top of the TV picture to the bottom. On High Def there are 1080 lines. To help visualize why the HD picture is better, think of the extra lines on HD as fitting in between the 525 lines on Std Def. With all those extra lines and the extra pixels on each line, there is more pieces of picture information, hence, the image is sharper.

Both of the images below currently look to be about the same sharpness. That is because the smaller the picture, the less you will see the difference between Std Def and High Def.

If you click on each of the images below and then make them full-screen, you will see that the HD image is sharper.

The same picture as it might appear on Standard Definition