what does it mean?
Closing the boy’s bedroom door, she checked the rest of the house, then went looking for his father. Beckett was right about Robert Strange. He drank and was undependable, an otherwise broken man who loved the boy as best he could. He worked part-time for a shade-tree garage far out in the county. The owner was a drunk, which meant Robert could drink, too. He worked off the books, mostly on American cars, mostly for cash. That’s where he would be, she thought, at the garage and useless and drunk.
What does "an otherwise broken man" mean?
What does " shade-tree garage far out in the county" mean?
What does "worked off the books, mostly on American cars, mostly for cash." mean?

Question

what does it mean?
Closing the boy’s bedroom door, she checked the rest of the house, then went looking for his father. Beckett was right about Robert Strange. He drank and was undependable, an otherwise broken man who loved the boy as best he could. He worked part-time for a shade-tree garage far out in the county. The owner was a drunk, which meant Robert could drink, too. He worked off the books, mostly on American cars, mostly for cash. That’s where he would be, she thought, at the garage and useless and drunk.  
What does "an otherwise broken man" mean?
What does " shade-tree garage far out in the county" mean?
What does "worked off the books, mostly on American cars, mostly for cash." mean?

Answer

* “Broken man” means exactly what you would expect it to mean. It is a person who had a difficult life, they are tired of their life, mentally overwhelmed.
* A “shade tree mechanic” is someone who works on their car on their own, without relying on professional repair shops. They are a DIY enthusiast, often taking unconventional and hacky approaches. By extension, a “shade tree garage” is a repair shop staffed with these untrained enthusiasts.
* “off the books” means that he was not paid officially, that is, he did not have an official salary, his earnings were not recorded in an account book, he was paid cash and probably did not pay taxes.

Answer

Worked off the books, mostly on American cars, mostly for cash means That the woman's husband used to work in a garage that was in the outskirts of their small town. The husband had no knowledge on how to repair cars. So he used to learn how to repair them reading from the books. The cars that he used to repair were made in America.

P. S. I may be wrong