The top of the cylinder, moving with the airflow, speeds up the air at any point compared with a non-rotating cylinder, while the bottom of the cylinder, moving against the airflow, slows the air down. Faster air corresponds to lower pressure, slower air to higher pressure, from Bernoulli along the streamline, and so it means there's a net lift force on the cylinder towards the top.
There's also drag pushing/pulling the cylinder in the direction of the flow.