From Diablo Wiki
Strength in damage and wield melee weapons as well as armor. Strength is one of the first vital attributes complemented with Dexterity to wield weapons and wear armor. Although Strength does allow the usage of heavier, stronger equipment, Strength only upgrades damage slightly.and is used to increase
This Strength attribute can be upgraded using a variety of spells, charms, armor and even weapons to accomodate such burden, but considering armor and weapons, those traits can only be activated if they are equipped. Once uneqquipped, the trait will not be in effect towards your strength. This is noticed by the blue highlight in your Strength Skill level, if white, no enchantments are effecting strength. If, for example you try to equip on a belt with the prerequisites of 110 Strength and you have 110 strength only by the equip of another belt, you would have to level up in able to wear your desired belt without the need of the enchantments from your previous belt. The same goes with any other armor or weapon. However, in the terms of charms, they are automatically activated as long as they are within your inventory. If placed in the stash, they will not activate.
Compared to Other Classes
In most cases, the Barbarians are the most effective by the use of the Strength Attributes. They need large proportions of their stat points consumed by strength in able to carry the largest weapons and deal the most damage. In opposite to that, a Necromancer or a Sorceress would be least likely to derive any such strength skills as they use more brain than brawn.
Other Interesting Characteristics
Strength may only slightly increase damage, but it increases damage for the two skill factors that you have selected on your screen. The Top indicator is your left selected skill, while the Bottom indicator is your right. If considering that a Barbarian was holding two weapons, the indicator would have two different damage mins and maxes for each weapon. Strength also increases damage for any future skills that you may use as default.