It's really hard! It's not possible for me to answer your question fully for every time and place. Some of this is related to histories that center objects rather than documents; this is not my area, so I will hope that others discuss that realm of history.

For histories based in documents, we have big problems looking at everyday life and we have even bigger problems researching people who lacked power (so for European and US history: lower class people, women, people of color, enslaved people, indigenous people, people with disabilities, and so forth). Because those groups, broadly speaking, had less access to money, education, and free time, they were less likely to have their lives recorded. And because archives were founded by people in power, the documents that have been preserved are less likely to document the pasts of marginalized groups. In most cases, to get at this information, which archives and documents almost purposefully hide from us, requires creativity, time, and dedication.

That being said, here are some ways historians get at the everyday life of the past:

Data projects: 15th Century Florence: There was a major tax survey done in Florence in the 1420s. That means that there are lots of documents relating to people in that city and their jobs. This data was all entered into computer databases by historians working in the 1960s (on punch cards!) and this painstaking work continues to pay off today. Lots and lots of historians have done work on this tax survey, which reveals to us intricate details about class structure and economy of Florence in the 15th century.

Reading documents 'against the grain': This is the term for trying to find information from documents that isn't the information they were intended to give. Usually the implication is that it involves finding information about oppressed people out of the documents of their oppressors. And often this is about European empires: empires require bureaucracy: there is tons and tons of documentation about European colonies. But that documentation is all from a European perspective, so trying to find the lives of oppressed people in those colonies is really hard. Nevertheless, people try to do it.

Finding documents preserved for other reasons (perhaps related to above): In The Book of Ages, historian Jill Lepore documents the biography of Jane Franklin, Benjamin Franklin's brother. Her letters were preserved in an attempt to know more about Benjamin, but they also reveal something about what it was like to be a relatively working class woman in the 18th century English colonies and early US.

For more recent events, historians can use oral histories.

I realize this is incomplete and I know these posts often disappear in a pile. I hope others also respond and give other, perhaps better examples.