To understand this, we have to first review the situation following the Peace of Augsburg. The resulting treaty was very much a compromise. (1) Each ruler determines the religion of his realm, either Catholic or Lutheran. (2) If a prelate of an ecclesiastic state was to convert from Catholicism, then he has to give up his post. (3) Due to his need for military support, Ferdinand added a third secret clause giving exemption to knights and some cities from religious uniformity in that realm, under some circumstances.
The first and second were the public articles of peace, and they are the reason why rulers cared about the beliefs of their people. The people are theoretically expected to follow the religion of their ruler, so if the ruler were to convert, then the people were supposed to follow.
A few examples perhaps served to illustrate the point.
First is the Koln War of 1538-1588. Gebhard Truchnsess von Waldburg the Archbishop-Elector of Koln wished to not only renounce priesthood but to convert to Calvinism and to marry. By the second article of Augsburg, he was legally expected to give or revert the archbishopric back to the Catholic church and to the Holy Roman Empire, however he instead wished to convert it to a hereditary realm. He may have had a point, because his family was very influential and were able to reliably use politics, influence, and money, to place their members on that seat and on other ecclesiastical seats. Further, as his subjects were overwhelmingly Catholic, his conversion would violate the first article of Augsburg. War broke out and expanded through intervention of belligerents of the 80YW in nearby Low Countries. In the end, the Imperial (Catholic) side won, setting the stage for the rise of the Wittelsbachs and a rise in Catholic power in the lower Rhine.
Second is the infamous revolt in Bohemia against their Catholic King, Ferdinand II the Holy Roman Emperor. Bohemia was largely Protestant, while Ferdinand was Catholic. However, for decades, Protestants were tolerated under a succession of weak or tolerant Habsburg emperors. But Ferdinand was a Catholic zealot, and rescinded an edict of tolerance signed by his predecessor Rudolph II. The people of Bohemia became angry and rebellion broke. Unwilling to be forced to convert and realizing that the Augsburg articles could force them to, they offered the crown to Frederick V, Elector of Palatine. Frederick's reign only lasted one winter, earning him the nickname "The Winter King".
I wrote a bit about how the 30YW itself grew to be a very large conflict here, where you can also find some sources and references.
In the run-up to the 30 Years War, how did religion get so politicized? Why did rulers care so much about the beliefs of their people?
I like the answer /u/Itsalrightwithme gave, but I can't help but think that it misses some important context and explanation that would more directly answer your question. I'm going to try not too make this too long and will also try not to repeat what's already been covered, but no promises. EDIT: I didn't succeed, sorry!
First, there's an underlying assumption to your question that should be addressed. Religion in the sixteenth and seventeen centuries did not "become" politicized—Christianity has been a political religion since at least the conversion of Constantine. Indeed, for most medieval and early modern Europeans, the idea of a "separation of church and state" would be awkward if not outright unthinkable. Our modern compartmentalization of "politics" and "religion" as two separate categories would make no sense to an early modern European; while the "Great Chain of Being" wasn't something that everyone believed (nor was it static), people commonly believed on some point on a hierarchy beginning with God and running through lesser supernatural beings to the Pope, monarchs, and then various lesser classes.
Of particular importance to the Thirty Years' War (TYW): the Catholic Church and most states were very closely entwined prior to the Reformation. Politically influential and wealthy people would have second or third sons placed in leading positions in the Church, the Papal States were an actual territorial force ruled by the Pope, and the Pope had significant influence on the policy and legitimacy of even relatively powerful monarchs. Most importantly for our discussion, princes drew their legitimacy and authority in part from the religious orthodoxy of their people; because most rulers believed that they ruled by the grace of God, heresy was considered a form of treason against the state as well as a sin.
The Protestant Reformation directly challenged the assumptions about the natural hierarchy and the relationship between the Church and the state. Many rulers, ranging from small German princes and cities to Henry VIII of England, saw the Reformation in part as a chance to reject Church authority and instead place themselves in control of a "national" or state-run Church. In many places, this meant a significant dismantling of the existing hierarchy. Rulers dissolved monasteries, seized church lands, rejected Papal authority, and enforced new rulings. Other monarchs saw these measures as evidence of greed and sinfulness, and saw Protestantism as a dangerous heresy that would undermine natural order and true faith.
This led to an awkward and extremely dangerous situation. Individual citizens of all classes, like rulers, were of very different minds about whether the Protestant Reformation (in all its flavors) were legitimate and true. But because Christianity and the state itself were so thoroughly connected, choosing a different faith than your sovereign was typically construed as a challenge to their authority. If you were Catholic under a Protestant ruler, authorities often suspected that your loyalties were to the Pope and not your country. If you were Protestant under a Catholic ruler, authorities suspected that you rejected your sovereign's authority alongside the Pope's. This led to persecution of varying severity across the parts of Europe affected by the Reformation: "Bloody Mary" burned hundreds of dissidents in England, while Catholic priests faced harsh penalties for performing Mass in many Protestant countries.
As time went on, circumstances forced Europe to admit that the divide between Protestantism and Catholicism would not be easily solved (as Holy Roman Emperor Charles V in particular had hoped before his abdication). This led first to the Peace of Augsburg after years of warfare, which as /u/Itsalrightwithme mentioned did not include the rapidly growing Calvinism. Calvinism upset Europe even further by causing dissension between Lutherans and Calvinists—much the same feelings as existed between Lutherans and Catholics also existed between Calvinists and Lutherans. Little trust existed between these groups (often called "confessions" in historiography), as they constantly sought to expand into different states. This led to "confessional politics", where distrust between confessions and a very unsteady, but fairly consistent, peace dominated European affairs up through the TYW. The Cologne War (1583-88) was the largest outbreak of direct conflict in the period but was nevertheless localized compared to the Schmalkaldic War before and certainly the TYW after. During the latter part of the period between those two wars, Protestants and Catholics each formed their own loose "leagues" of military alliance in preparation for a conflict many believed inevitable.
Additionally, despite the intentions of the Peace of Augsburg and most princes, rulers of one confession very often had populations of different confessions. Most importantly for the TYW, the nobility of Austria and Bohemia were overwhelming non-Catholic despite both being ruled by the Catholic Habsburgs. Habsburg efforts to change this were largely successful in Austria during the so-called "Counter-Reformation" that saw Catholicism restored to many places were Protestantism had initially spread; this caused a panic among Protestants everywhere, but particularly in the Holy Roman Empire (and especially Bohemia).
This leads us up to the commencement of the TYW. When the previous Austrian Habsburg ruler and Holy Roman Emperor died, the Bohemians, who had an elective monarchy, saw a chance to select a prince who followed their faith (which was largely a mix of Calvinists and Utraquists/Hussites). However, they had already promised to elect Ferdinand II of Austria, who was also easily elected Holy Roman Emperor. The monarch chosen by the Bohemians in 1618, after a couple others refused consideration, was young Elector Palatine Frederick V, a Calvinist and leader of the Protestant League. Frederick appears to have believed, though to what extent is debated, that he was playing a central role in a conflict that would determine the fate of Christendom and possibly even lead to Armageddon. Ferdinand II, for his part, believed that the re-establishment of Catholicism in his lands was a foremost duty, and many believed that this campaign would also include the entire Holy Roman Empire and not just the lands he directly ruled.
tl;dr Politics was absolute inextricable from religious considerations in early modern Europe. Most rulers saw other confessions as a threat to their authority, both in terms of legitimacy but also often simply a physical threat. In short, religion did not "get so" politicized, it already was politicized and the Protestant Reformation made this politicization into an enormous threat to the stability of Europe both diplomatically and within individual states. The Thirty Years' War was the seemingly inevitable result of the tension between the three major confessions.
Sources:
Thomas Brady, German Histories in the Age of Reformations, 1400–1650
Peter Wilson, The Thirty Years War: Europe's Tragedy
Brennan Pursell, The Winter King: Frederick V of the Palatinate and the Coming of the Thirty Years War