How did Russia's revolution garner support enough to topple the Czar's government?
Russia's revolution was able to do this because there was so much popular unhappiness with the Czar and his government.
The Czar's government had been unpopular for decades. There had been attempts at revolutions well before 1917. The government was very conservative and unwilling to share power or to do much to improve the lives of the lower classes.
This discontent was magnified by the events of WWI. The Russian military did very poorly in this war. This led to discontent and even mutinies among the military by 1917. When this started to happen, it was like a catalyst for all the discontent that had been brewing.
Russia's revolution got support, then, because the Czar's government had long been oppressive and because of the shock of how badly it did in WWI.