The people of Lystra spoke Greek and Latin. It is likely that Paul spoke to them in Greek, but the people of Lystra responded to the healing of the man who had been lame since birth in their local Anatolian language, that Luke calls Lycaonian. Barnabas and Paul may not have understood at
first what the people where saying, but when the priests of Zeus showed up at the gate with
a bull decorated in wreaths, they finally interpreted the people to be saying:

“The gods have come down to us in human form!”

The Roman poet Ovid (43 BC-17 AD) had written a poem about Zeus and Hermes that was circulating in this part of the empire during the first century (more here; The poem is here; Literal  English translation with parallel Latin here) In the poem Zeus and Hermes disguised themselves as an old man and woman and traveled unrecognized though the land around Lystra called Phrygia to a thousand homes before they were welcomed and cared for. 

The worship of the Greek gods Hermes (called Mercury by Rome) and Zeus (called Jupiter by Rome) in this area of Asia is confirmed by archaeological discoveries that uncovered inscriptions and stone carved images that portray these two gods. 

The identification of Barnabas as Zeus and Paul as Hermes most likely comes from the fact that
Paul was doing the speaking, because Hermes was the messenger god for the chief god, Zeus.

The people of Lystra wanted to honor “their gods” for the miraculous healing of a man who had been lame since birth. The lame man’s muscle tissue and bone mass had never before been developed
or used for standing or walking. But, when the lame man heard the good news of Jesus, Paul saw
he had understood who Jesus was and believed the message to be true. At Paul’s command the
man responded to the in faith by rising to his feet, and as he rose he miraculously gained the
muscle tissue and the bone structure to stand, and then, to walk for the first time in his life.