/u/ninjahund

To start with, it's important to distinguish between Christian IV and the council of state (Rigsraad) representing mostly the aristocratic elite of the kingdom of Denmark; they were in a power-sharing agreement. In the 1620s, the Rigsraad was concerned by two developments: the war in Germany, and the situation in the Baltic. The war in Germany I hope you already know a little about, what with Catholic victories ushering an era of worry for Protestant powers such as Denmark.

Territorial ambition on land and sea

At this point in history, the situation in the Baltic is a long evolution of power struggle between Denmark, Sweden, Germany, and Poland. Denmark wished to preserve their Dominum maris Baltici as it was a very important source of revenue. Further, the elites of the aristocracy preferred peace at all cost, both with Sweden and with Germany (HRE).

Christian IV harbored some ambition to be the leader of Protestant powers, against the wishes of the council. He had a strong finance behind him, he had a surplus of 250,000 thalers per annum, and a further 200,000 thalers from tolls levied at Elsinore (Helsingør). Finally, thanks to victory of Christian IV over Charles IX of Sweden in 1613, Sweden had to pay annual indemnity to Denmark.

But just like any other strong ruler of the era, Christian IV sought more. In particular, he saw certain territories in the Lower Saxon Circle as alluring targets. He wanted them to be used to extend his sphere of influence beyond his capacity as Duke of Holstein, and to provide positions for his younger sons. For that, he coveted principalities (former bishoprics) of Bremen, Verden, and Osnabruck. If he were to succeed, he would have control over key river estuaries of Else and Weser, further strengthening his domination over that side of the Baltic.

He largely succeeded by 1623-24, finding himself in a strong political, financial, and military position as the forces of the Catholic League was being disbanded following their success against the Elector Palatine and Ernst von Mansfeld.

Political ambition turns the tide of war, or political reality forces a change in policy

At this critical juncture, other Protestant and anti-Imperial powers increased their interference in German politics. The French had lobbied for Bavaria to break against the Imperial army, but failed. The French approached their former ally Denmark but Christian IV and his council refused to continue its war. After all, Christian had succeeded in his territorial goals. On the other hand, Gustaf Adolf II of Sweden was eager to extricate himself from the war against Poland, so he signed a six-year truce where he conceded territorial control but earned the very-important toll revenue of all ships calling at ports of Poland and Prussia. This ended up being a third of Sweden's revenues. So as she prepared to enter the 30YW, Sweden had an experienced body of troops, a well-organized state, and intended to drive down the Vistula to Bohemia. The condition being, Gustav Adolf II was to be commander of a coalition army which limited direct interference from France. Essentially, Gustav insisted on strategic leadership.

Seeing this, Christian IV feared that if Gustav Adolf II -- viewed as a rival -- were to succeed, Sweden would be the leader of a Protestant Coalition based around northern Germany and the Baltics, essentially a threat to his recent territorial gains. This is why Christian IV entered the 30YW in 1625. He essentially jumped the gun against perceived threat that more recently is viewed as very real threats in order that he could claim leadership of the Protestant Union. In terms of historiography, there was tendency to label Christian's decision as simply foolhardy over-extension and over-confidence. This is no longer the prevailing view, especially considering how things looked like in 1625: the Catholic League were largely immobile, and Protestant forces on the other hand seemed ready for action. It looked to both Christian and Gustav that all the Protestants needed was a leader to lead them to victory.

The best read on this subject is Parker's The Thirty Years' War, in the chapter titled "The Danish Intermezzo".