I don't think there's a definitive answer, as different people most likely had different reasons for converting. But Swedish historian Dick Harrison (in History of Sweden 600-1350) states the following:

• While missionaries had come to Scandinavia for at least centuries before, and there had been prominent members of society who were christian, it really took off around the start of the 11th century when a few kings converted to and promoted christianity. He states in passing that they forbid other religions to an extent. This established their authority as derived from God (rex Gratia Dei), and was part of establishing the office of king as an institution rather than being tied to a particular person (this was a gradual process over generations). He does not state this as a reason for conversion however, only an effect.
• He notes that archeological signs point to rich/prominent women being among the first to flaunt their religion (such as raising runestones proclaiming their faith) in the centuries before. This does not necessarily mean they were the driving force however.
• The kings were not alone in converting the populace, but took help from and instituted priests to help them. Priests and bishops were assigned by the local authorities rather than the Pope.
• It's often hard to tell, when finding christian items from the time, to determine wether they imply that someone was christian or simply had it for non-religious purposes (like jewelry for example). Not a lot of texts on the subject remain from the time period, so this is based more in archaeology than from written sources.

Disclaimer: I'm not a historian, I'm only summarizing the chapter on the conversion from the book (which is in Swedish, I don't know if it's been translated).