I read that Japan, Italy and Germany emerged as powerful nations around 1870. What led to this?
To the extent that the emergence of these three nations was all caused by the same things, they were caused by the Industrial Revolution and the rise of nationalism.
The Industrial Revolution changed the nature of military power in the world. Military power came to be embodied by steam ships and cannons. This was especially important in the "opening" of Japan and its rise to power as it industrialized to keep up with the West.
Nationalism also brought about the rise of these new powers. Ethnic Germans and Italians came to feel that they should coalesce into one country instead of being split into many separate states. This allowed them to create new nation-states that had more potential for power.
These two major trends of the 1870s, industrialization and the rise of nationalism, generally contributed to the rise of these new nations.