Well, you have to understand that the colonists were facing what they believed to be an implacable foe, evidenced by the numerous attempts at simple change. The Olive Branch Petition of 1775, was sent to the King to attempt to express grievances and find peace. To what extent this really was an attempt at peace depends who you ask, but the long and short of it is this: it was de facto rejected and the King had already proclaimed the colonists traitors.

The King, for his part, had already issued the Proclamation of Rebellion in August of 1775, and Massachusetts had been declared in a state of rebellion in February of that year. It's not that the colonists wanted to necessarily split off; the Olive Branch Petition does continuously mention in some sense that the colonies would like to retain their status, merely enhance it to protect their interests better and end hostilities. An example:

Attached to your Majesty’s person, family, and Government, with all devotion that principle and affection can inspire; connected with Great Britain by the strongest ties that can unite societies, and deploring every event that tends in any degree to weaken them, we solemnly assure your Majesty, that we not only most ardently desire the former harmony between her and these Colonies may be restored, but that a concord may be established between them upon so firm a basis as to perpetuate its blessings, uninterrupted by any future dissensions, to succeeding generations in both countries, and to transmit your Majesty’s name to posterity adorned with that signal and lasting glory that has attended the memory of those illustrious personages, whose virtues and abilities have extricated states from dangerous convulsions, and by securing the happiness to others, have erected the most noble and durable monuments to their own fame.

However, as the war dragged on and peace eluded both sides, the intransigence and continuous escalation of hostilities meant that the Americans were no longer willing to submit themselves once more to the monarchy. The statement "Give me liberty or give me death" was not just a proclamation of the fervent nature of the American want for freedom; it was very plausible that death would come if they could not secure liberty. The Proclamation Rebellion, and the way it promised to punish those responsible, meant that many of the men fighting were in a fight for their lives, and couldn't risk any reconciliation given the bad blood shared already by the time of the Declaration of Independence.

And that's just one take on it :).