Light Radius is the illuminated circle around players that allows them to better see their surroundings.
Some items can change the Light Radius of the player by equipping them. Charms do not require the player to equip them in order to increase the Light Radius.
Improving light radius through the magical properties in items means that a character can more accurately see enemies and therefore can avoid surprises and fire from afar when required more easily.
Diablo I
In the original Diablo, Light Radius also had a tactical aspect, where it would affect when monsters took notice of the player and would begin advancing toward them to attack them. In this way, characters with a large light radius could trigger monsters from far off and save some walking, whereas ones with a small light radius could sneak past groups of them.
Diablo II
In Diablo II, the base Light Radius is 13 and the max is 18. So there is no point in a character's equipment netting more than +5 to Light Radius.
Diablo III
In Diablo III, Light Radius stat is non-existent. Most areas are brightly lit, and in darker ones (and in the night) characters have a static radius of lighting that cannot be changed by any means, and does not affect the gameplay in any way.