This reads like the difficult English used by academics, in fields like literary criticism.
"Trace" here means "a mark, or something else that is left behind, by something that has gone away." For example, "the medical examiner found traces of drugs in the dead man's body." Or, "the stone walls in these woods are the traces of former farms."
In this case, the "trace" is the latent image captured in the film.
To "inscribe" simply means "to write," and "inscription" means "writing." The "mode of inscription" means the way in which the image is captured in the film. Here, the mode of inscription is a chemical change that light causes in silver salts in the film. (Notice that the "-graphy" root in "photography" also means "writing." A photograph is an inscription made by light).
The "image depicted" is the real-life scene outside the camera, which the camera "depicts." The "trace" is the corresponding image recorded by the film. The "mode of inscription" is a chemical change caused by light. The "physical connection" is made by light energy, focussed onto the film by the lens.