The basic reasons why trade prospered in Central Asia (and beyond) under Mongol rule are that the Mongols controlled a vast area of land and that they very much wanted and encouraged trade. As Bernstein says on p. 90 of the book,
…the overland route from China to the gates of Europe lay in the hands of a more or less stable chain of Mongol states, which enthusiastically embraced commerce…
There are two aspects to this statement that we should discuss further. First, it is important the Mongols controlled all of this territory. When one empire controls a vast area, trade is typically easier. The empire can keep the peace within its boundaries and can make trade safer. The fact that there is one empire means that traders can go a long distance without having to deal with different governments that have different rulers. These things make trade easier.
Second, it is important that the Mongols wanted trade. The Mongol homeland did not possess many natural resources. The Mongols themselves were nomadic herders. Because of this, they wanted to bring in goods and technology from other places to enrich and empower themselves. They felt that trade was important to their survival and so they encouraged it.
For these reasons, trade prospered during the time of Mongol rule.