This page of the Metro Exodus Sam's Story guide provides step-by-step instructions on how to get the Cinephile achievement.
First you need to get to the building presented in the map above. You can easily recognize this place by the loud music. Inside, you will find a projector that plays an old movie.
The recording room is right next to the projector. Go to it and press the interaction button. Sam will start searching for records - he picks up the tape with the US flag.
Go back to the projector and play the tape. You will see footage of one of the US cities. This way you will get the Cinephile trophy.