Choose the meaning of the underlined phrase according to the context of the passage:
One day he took his flute and a fish basket, jumped on a rock and began to play his time.

Play for time (Phrase)
English Meaning: to try to make something happen later instead of sooner
Synonym: stall, temporize, gain time, hang back, procrastinate, beat about the bush, drag one's feet, delay
Example Sentence: He played for time by establishing an advisory committee.

Source: Oxford Dictionary, Merriam-Webster Dictionary

As per the context of the passage, in order to catch fish, the fisherman climbed on the rock and started playing his flute slowly and calmly because fishing is a time-consuming process and it requires time and patience.