In order to understand this passage, you need to understand the meanings of three words. These words are “divest,” “prejudice,” and “prepossession.”
To divest yourself of something is to get rid of it. In the 1980s, people put pressure on colleges to divest themselves of stocks in companies that did business in South Africa (because of the apartheid regime). We do not use this phrasing much today, but you can divest yourself of a piece of clothing. The phrase just means to make it so you no longer have the thing.
Prejudice and prepossession are synonyms. They both mean ideas or judgments that you have about something or someone before you have really thought deeply about the thing or before you have gotten to know the person. We typically use this in racial terms, saying that someone is prejudiced against members of a given race because they have a bad image of those people without having gotten to know them.
Given these definitions, you should be able to see what this passage means. In Common Sense, Paine is trying to convince people that monarchy is wrong and that the American colonies should become independent. He is asking people to get rid of the ideas and judgements they may already have on this issue. He wants them to think about it deeply and clearly. He believes that, if they do so, they will come to agree with him.