When and why are the pilgrims traveling, and how does the host quickly win their trust?

The first line of the general prologue begins (in modern English), "When, in April,...", so the answer to your first question about when the pilgrims are traveling is "April". They are headed to Canterbury to worship at the shrine of St. Thomas a' Becket.  This was a common pilgrimage in the middle ages, one looked upon, by many, as a vacation of sorts. The host, Harry Bailly, wins approval and trust from the pilgrims by being a gracious host.  He gives each one a seat at the Tabard Inn as he/she arrives and gives them all good food and drink.  He offers to come along as a guide.  He is genial, likes to joke, and is good at organizing people.  He suggests that to help pass the time along the way to and from Canterbury, each pilgrim tell tales. Of course, Chaucer died before he finished writing The Canterbury Tales, so we do not get all of the pilgrims' tales.