In migration studies one usually speaks of the "push and pull" factors -- the things that motivate you to leave, and the allure the destination has.
In the case of the United States (for Germans in the 19th century) there were distinct political and socio-economic factors on both the push and pull side:
- poverty and nutrition: German speaking lands were experiencing a fairly strong population growth during the 19th century, which had to do with the beginning industrialization, which was however lagging for Germany in comparison to France and the United States; as a side effect, impoverished land population moving to urban regions were not meeting the expected employment they were hoping for. E.g. a Prussian government district like Trier in the 1830s had as much as 30% of its young men emigrate to the United States (which might have made Karl Marx an American if things had played out differently) for hope of a (better fed) future.
- religious tension: While there were no longer religious wars in the Holy Roman Empire or German speaking lands after 1648, there were still local tensions between protestant minority groups and Catholics or Protestant Majority Lutherans, the latter being less tolerant of other Protestants than one might assume. Thus, smaller enclaves of religious groups in larger sections of a dominant other were indirectly encouraged to look for places to move to as they were facing daily "glass ceilings" in dealing with their neighbours*;* this affected, in particular, smaller protestant groups in the German South, who were often stuck in the middle of largely Catholic reactionary areas, and moving to the more Lutheran north was not always a viable alternative. States like Bavaria had "banned" protestantism still as late as 1810. While the middle of the century saw somewhat a drop in religion-based migration, the Kulturkampf between Catholics and the Protestant-dominated German empire after 1871 saw a rise again of religion as a migration factor.
- politics: German pro-democratic activists were fighting for democratization where they could, but often ran into harsh oppressive local governments in the process; many of these activists, over the course of the 19th century, eventually fled to the United States. The advent of steamships crossing the Atlantic after 1830 became technically a catalyst for migration, as both cost and danger of emigration suddenly dropped, while oppression slowly increased. During the 1848 revolution in Germany, it is noteworthy that emigration drastically dropped as the political outlook for a democratized united Germany suddenly seemed good; but after the revolution was crushed in 1849, a massive emigration wave for the United States picked back up. Many of the German democratic activists of the 1848 revolution can be found a few years later during the American Civil War on the side of the Union, fighting as abolitionists. After German unification and the forming of the German Empire in 1871, overall emigration numbers dropped somewhat as the new Reich constitution somewhat cleared the field for more personal liberty, even if politics remained fairly reactionary as a whole: elections were now enshrined in the constitution, the economy had picked up, and there seemed to be a political way forward to liberalization without violent revolution.

A fairly "classical" German migrant who came to the US in 1848-1852 is, fairly clearly, someone who had supported the democratic revolution and felt the need to leave quickly for fear of repercussions. Earlier German migrants are more likely to have religious and/or economic reasons (though the period 1830-1848 had still a large number of people looking for political refuge), as do those after the 1870s. But these are broad brush strokes, and the individual mileage varied.