Main-belt asteroid (6478) Gault unexpectedly sprouted two tails in late 2018 and early 2019, identifying it as a new active asteroid. Here we present observations obtained by the 1.2-m Zwicky Transient Facility survey telescope that provide detailed time-series coverage of the onset and evolution of Gaults activity. Gault exhibited two brightening events, with the first one starting on 2018 Oct. 18$pm5$ days and a second one starting on 2018 Dec. 24$pm1$ days. The amounts of mass released are $2times10^7$ kg and $1times10^6$ kg, respectively. Based on photometric measurements, each event persisted for about a month. Gaults color has not changed appreciably over time, with a pre-outburst color of $g_mathrm{PS1}-r_mathrm{PS1}=0.50pm0.04$ and $g_mathrm{PS1}-r_mathrm{PS1}=0.46pm0.04$ during the two outbursts. Simulations of dust dynamics shows that the ejecta consists of dust grains of up to 10 $mu$m in size that are ejected at low velocities below $1~mathrm{m~s^{-1}}$ regardless of particle sizes. This is consistent with non-sublimation-driven ejection events. The size distribution of the dust exhibits a broken power-law, with particles at 10--20 $mu$m following a power-law of $-2.5$ to $-3.0$, while larger particles follow a steeper slope of $-4.0$. The derived properties can be explained by either rotational excitation of the nucleus or a merger of a near-contact binary, with the latter scenario to be statistically more likely.
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