"To press in" is similar to "to crowd". People can press in on you and can physically touch you and push you. But eyes and faces can also press in. In this case, it is not physical (actual touching) but is more like staring or glaring. So if faces press in on you, they are mostly getting very close to you and are staring. They are crowding you and usually making you feel uncomfortable.

Can you give some more context? Either way, it means that people are crowding around and staring at the narrator, but I could probably explain it better if I had more context.