What does this phrase mean?

This means that the conductors of the experiment had direct control over this factor of the experiment and could vary it in order to see how it affected outcomes. In this case, that variable was how intimate the contact was between the experimental subject of the experiment and the “authority figure”, the person instructing the subject to administer pain. The experimenters wanted to see if having the authority figure right there beside the subject giving instructions, as opposed to phoning in instructions, was a factor in the subject’s behavior.

Here’s a description of variations in the experiment that explored the affects of different levels of immediacy (this demonstrates pretty well what your phrase means):

“The participant’s compliance also decreased when the authority’s physical immediacy decreased (Experiments 1–4). For example, in Experiment 2, where participants received telephonic instructions from the experimenter, compliance decreased to 21 percent. Interestingly, some participants deceived the experimenter by pretending to continue the experiment. In the variation where the “learner’s” physical immediacy was closest, where participants had to physically hold the “learner’s” arm onto a shock plate, compliance decreased. Under that condition, 30 percent of participants completed the experiment.” (source)

The way I read it is—
1. They did an experiment and named it the “Stanley Milgram Experiment”
2. All experiments have variables (or factors) that can influence the result of an experiment.
3. Some variables can be adjusted, some can’t. In this case, they had a variable that could be adjusted.
4. The variable they changed was how far away from/how close the authority figure was to the subject—in the room, outside the room, etc.
5. “Degree of physical immediacy of an authority” describes how much distance there was, physically, between the test subject and the authority figure telling them what to do.
6. The results of the experiment were influenced by how much physical presence the authority had in relation to the test subject.

Visualize a graph with an x and y axis. The independent variable would be laid out on the x axis and the dependent variable (usually things like race, age, gender, seemingly ‘static’ characteristics) is laid out on the y axis. Throughout the above experiment, the independent variable or, more specifically, proximity to physical authority was the element varied to see what effect there would be on the subject.

I don’t see anything wrong with the word “proximity” instead of “immediacy”.

@flo I think I see why the distinction is important. “Immediacy” carries the sense of how mediated the interaction is. For example, I could be standing three feet away from you and talking to you with nothing between us; or I could be standing three feet away from you but we could still be in separate rooms and communicating by telephone. The proximity would be the same, but in the second example the communication is mediated by the intervention of a wall and a telephone. It’s less immediate.

@thorninmud that makes sense.