It means to search for multiple meanings to some words.
Explanation:
When people say to read between the lines, you have to look closely at the text. You have to think differently about meanings of words and sentences.
For example: "Johnny grinned. He passed the test! He laughed, but then he cringed, wincing."
This could be taken in different ways. You could say that he was happy he passed the test but because he was injured, he cringed when he laughed. However, you could also say that he laughed because he was happy he passed the test but he cringed because he remembered his grandma just died and he shouldn't be happy.
