Using AC current to trigger Triac
If the load current is extremely tiny then the triac may indeed fail to trigger. This is quite intentional, and the 1K resistor helps. 

When the load resistance is low enough the current through the triac gate will exceed the trigger current and the triac will turn on near the zero crossing (see the MOC3043 datasheet for what 'near' means). Note that the current will flow primarily through the triac gate before it triggers and when that current exceeds about 1mA (due to the 1K resistor). After it triggers, the current flows through the triac. 

Again, if the load resistance is low enough, the current through the triac will rise to a level that exceeds the holding current (see the triac data sheet) during the brief zero-crossing interval established by the MOC3043. 

Only if both those conditions are true will the triac switch on properly for the entire AC half-cycle. 

Suppose that the holding current of the triac is 50mA and the trigger current is 5mA, and that the zero crossing voltage is 15V (just picking some numbers). For the triac to trigger, the load resistance must be less than about 2K\$\Omega\$, and for it to stay on it must be less than about 280\$\Omega\$, which is about a 50W load on 115VAC.