Why is thick steel wire stronger than thin steel wire?

The material will yield when stress reaches a critical value.

Stress = Load / Area

Thick steel wire is stronger than thin steel wire because there is more cross sectional area in the thick wire. Although the material's strength in load per unit area would be the same, the ultimate load that the wire can sustain would be more in the thick wire. A simple way of looking at it is to imagine a thick wire as a number of thin wires stuck together. If a thin wire can support a mass of 1kg then 2 thin wires can support 2kg. A wire which is twice as thick (twice the cross sectional area) can also support 2kg.