There were a myriad of things going on in the 1840s and 1850s that would have compelled a German person to leave for America. For clarity sake if I say Germany, I mean any of the German states that would come to compose Germany proper. If you know where your ancestors were from I could give you more specific information about why they may have left. Push factors for migrants in this time included difference in religion, war, politics, and money. The same things that drive immigration everywhere really, but in Germany there was a convergence of all of these factors at once. Pull factors towards the United States in particular included lots of open land, freedom, and cheap quick travel.

Religion: Many German states had official state religions. Some of the states were Catholic, and some were Protestant. If you weren't Catholic, or the right brand of Protestantism, you faced discrimination. This wasn't discrimination like you might think of much later in Germany where you would be killed or tortured, but instead softer discrimination like being denied certain permits, or not being allowed to attend particular schools or to get a government job. Germans might also be required to tithe money to a church they do not personally subscribe to. Their church might be shut down. Many groups, especially Free Thinkers, Catholics, and Anabaptist and Mennonites left due to this kind of religious persecution, or to seek greater religious freedom elsewhere. Just moving within Germany was sometimes not enough. Catholics left in larger numbers as Prussian hegemony extended across all of Germany, and they suffered some persecution almost everywhere. German Catholics and Free Thinkers were important to the rapid growth in American intellectualism during the 1850s, and the growth of the American education system.

War: The German states were undergoing political turmoil in the late 1840s, that culminated with the Revolutions of 1848, which lasted through about 1850. The revolutions attempted to replace many of the kings and princes with republics or constitutional monarchies, depending on the faction. The revolutions were all eventually crushed, largely due to disorganization, and ushered in strong backlash against those who had participated or sympathized in them. Even being accused of association with one of the liberal groups could result in imprisonment or blacklisting, both of which make making a living harder. Many of these revolutionaries who fled for the United States would form the backbone of the abolitionist movement prior to the American Civil War, and enlisted by the tens of thousands for the Union army.

Economics: Germany underwent a slow centralization in the mid 19th century, culminating with the German unification in 1871. As the nation centralized, greater industrialization and urbanization followed. This resulted in wages dropping for artisans and farmers due to increased efficiency and throughput in factory work or centralized farms. Growing urban areas wanted more luxury goods, meaning more farmland was dedicated to beef and pigs and less towards crops like grain. To make up for the overall caloric deficit, farmers came to rely on potatoes more and more for sustenance. When the potato blight swept through Europe, many people faced hardship due to the destruction of crops, and rising prices on grain meant the average person could afford food. The blight was nowhere near as bad as in Ireland, few to no people actually died of starvation as a result of the blight alone, but it made living harder for the poor. Many of these displaced farmers and craftsmen would settle in the United States, giving birth to the American brewing industry and the rapid growth of the breadbasket in the midwest.

Lastly, some German states participated in sponsored immigration. The local governments realized it was cheaper to get rid of out of work people rather than pay for their social safety net, so they paid for them to permanently relocate to the United States, Brasil, or Australia. Many German states had welfare systems like state health insurance or unemployment benefits. The trip overseas was very cheap comparatively, and quick and safe enough many people volunteered at the chance. It also subsidized German passenger shipping and helped them compete with already established British shipping interests.

I hope that helps clarify it some for you!

Edited for clarity.