Why wouldn't you? I guess from your question you're assuming that the average agricultural labourer in a fairly typical Northern European agricultural society had time to really consider the what we now regard as the obvious implications of the dispute between Parliament and the King.

Factor in a couple of other possibilities. Lets say you are a tenant farmer, not the absolute bottom of societal structure but the majority of people in the society are like you. Most of the country is run via the manorial system, that is - a local landowner from whom you rent lands to farm. This is your world, it is mostly unchanged for hundreds of years, it is essentially medieval. You may be fairly well off, you may be struggling, but generally you are god fearing, hard working and thankful to be in an England that is free from war and famine. You were taught, and teach your children that this order they see around then - in contrast to barbarous or war torn lands, is because of a hierarchy. God is at the top, the King is his agent on Earth, and then the Kings family, his relatives, his ennobled representatives, knights, gentlemen, and on down to you. This divinely sanctioned order is pleasing to you, because it reflects the way you see yourself. As a free man and master of your house, you are at the top of your own little hierarchy, which includes your wife, family and servants. One day even your youngest child, if God is good and you pray daily, will have his own household and his own hierarchy to manage, free from severe want, pestilence or war.

Now, skip forward to the civil war. The civil war starts as a series of disputes between Parliament and the King. As an onlooker however, you aren't going to know the details, only that there is dissent and rebellion against the King. Remember the hierarchy. Would you tolerate your servants if they decided to stop serving you? No, you would not. So the way the King is being treated threatens the divine order of things. Philosophical disputes about justice and freedom haven't come into it yet, and to be honest they won't get spread about until the war starts to get really bitter and brutal. The civil war starts in many people's minds as an very familiar, almost medieval kind of dispute. The King's evil power hungry councillors have plotted treason and open rebellion against him. You must do your duty as the Kings loyal subject to put down this insurrection. You must also prevent the disorder and strife in the realm that could hurt your family or see your lands ravaged by thieving rebels.

That's very brief. You must also add in a whole host of other factors, a huge one is religious division, which moist people were keenly aware of in those days. Religion became a massive factor especially when Charles enlisted the help of Catholics, this turned a lot of ordinary folk against him, because they had been taught papacy was a gross threat to the natural order of England (remember Guy Fawkes) and was essentially foreign in nature.