Why did Paul stay with Aquila and Priscilla in Corinth?

They shared the same trade, tentmaking, so he abode with them and worked. Aquila and Priscilla had recently come from Italy to Corinth after Emperor Claudius commanded all Jews to depart from Rome, and Paul, arriving from Athens, joined them in their occupation.

Les explains that Paul had departed from Athens and came to Corinth, a bustling port city with two ports—one west to Rome and one east to the Mediterranean—commercial, and grossly immoral. Paul came with the gospel of grace to the Gentiles. In Corinth he found Aquila (called Achela in the transcript), a Jew born in Pontus, lately come from Italy with his wife Priscilla, because Claudius the emperor had commanded all Jews to depart from Rome. Paul came unto them, and because he was of the same craft or trade, he abode with them and worked. For by their occupation they were all three tent makers. Les notes there is some controversy among scholars about what the tent was.

Acts 18:3

And because he was of the same craft, he abode with them, and wrought: for by their occupation they were tentmakers.

What occupation did Paul share with Aquila and Priscilla?

Paul shared the occupation of tent making with Aquila and Priscilla. Acts 18:3 records that Paul was of the same craft as them and worked with them, for by their occupation they were tent makers. There is scholarly debate about the exact nature of the tents.