I'm using this triac driver to control a triac. In its datasheet it says 'provide random phase control of high current triacs [...]'. What does that mean? It doesn't have zero-crossing detection?
Yes, it means it doesn't have zero-crossing detection. The latter is useful for purely resistive loads, where the current is in phase with the voltage. For reactive loads, inductive or capacitive, this doesn't offer an advantage.
"Random" just means the triac switches on at the time you signal it to switch on, whatever the voltage phase at that moment. So its actual meaning is that it can switch on at any time.
A note I made earlier on zero-crossing switching:
You may have noticed that incandescent bulbs always fail when they're switched on. That's because the mains phase can be near its maximum when switching on. Combined with the low resistance of a cold bulb this results in a high current peak, which may burn the filament. When you switch on a zero crossing you avoid these peaks.
Random switching is needed if you want to build a triac-controlled dimmer, where you want control of the phase angle where you ignite the triac over the full 180° of a half cycle.
"Random phase" is just a marketing term for, as you suggest, "hasn't got the sophistication of zero crossing detection".
A random-phase driver will provide a trigger signal whenever the input trigger conditions are met. Mains voltage can be at any point in its cycle and will trigger immediately. With a TRIAC the device will remain turned on until the next zero crossing point is reached - about 0 to 180 electrical degrees later.
Random phase has the advantage that controlled voltage can at least notionally be varied more finely that zero crossing triggering allows. Zero crossing only allows whole half cycles so on periods are in multiples of 10 ms (50 Hz mains) or 8.33 ms (60 Hz mains). In most cases this is not an especially important advantage.
Zero crossing triggering minimizes RFI, notionally stops inductive transients and will place less loads on e.g. lamp filaments. With highly reactive loads (inductive or capacitive) current and voltage are not in phase so zero crossing switching still leaves problems to be solved.