It means they were hitchhiking. When you're trying to get a free ride from someone, you stand along side the road and extend your arm out to your side with your thumb upraised. (A "lift" is slang for a ride.)
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In my early teens I ran away from home, for the simple reason that I was bursting with curiosity about the world outsite Brooklyn. I talked a pal of mine into going with me. Our thumbs got us lifts, we sang for food, and at night we appeared at the local police station to announce we were hitchhiking to relatives and asked to be put in a cell until morning. It worked well until we reached a small town in Delaware.
Could anyone explain the highlighted sentence please?
Yet, please allow me this chance to sharpen my dull pencil. I'll try to use simple words to rephrase the lines in pink.
We raised our thumbs to get our free ride everywhere, we sang songs (in joints or restaurants) to exchange for our meals, and at night we got to the local police station to tell the police that were thumbing for free ride to visit our relatives and asked to stay in a small room there until next morning.