Why are the church leaders in Antioch seeking God through corporate prayer and fasting?
The church special time of seeking the Lord together. It isn't a human-focused "revival" to save souls or return backsliders; it is focused on Jesus, on drawing close to him and listening to him. Biblical knowledge of God is not enough the church need a practical knowledge which only come through fellowship and communication by prayers.
How does God respond to their seeking ?
God respond by speakings to them via the prophet and prophetess. He gave the church directives on who to send for mission and who to keep. Thus He directed "Seperet me Paul and Banabase for the work I call them"
Why do you think we don't do more of this sort of thing today?
We don't do more of this sort of thing because of self wisdom that have taken over the worship of Christ today, Again lack of knowledge of the apostolic doctrine and method. Finally may have refuse mentorship by good mentor. Only a man properly discipled can know this. Like me despite my many years in the Lord this my first time hearing this.
What was the importance of this to the church at large?
For growth. For successful mission and for good representation of Christ this communion with God by fasting and praying is very important.