It seems that stock index futures affect economic growth, but I don't know how they work. Please can someone explain? Thanks in advance~
Actually empirically there is causality going both ways there (i.e. economic growth helps stock index futures to grow but also stock index futures help economy to grow). However, focusing only on the effect going from futures to the economy, the reason for this relationship is the fact that capital markets (including the futures market), help to ensure efficient allocation of capital and this empirically has first order effect on economic growth (e.g. see Levine, 2005).
As summarized by Şendeniz-Yüncü, Akdeniz, & Aydoğan (2018):
... well-functioning futures markets allow for greater and more efficient risk sharing, thereby making it possible for firms to undertake relatively riskier projects and, hence, promote growth. On the other hand, futures markets widely distribute equilibrium prices that reflect demand and supply conditions, and knowledge of those prices allows investors, consumers, and producers to make informed decisions. Consequently, the amelioration of information and transaction costs fosters efficient resource allocation, thus leading to economic growth.
In addition note that literature overall suggest that the effect from the direction of stock futures gets weaker for high income economies. This is thought to be due to the fact that capital in high income economies is already more or less efficiently allocated.