How does a receiving antenna generate current/voltage?
For reference, check out books by Balanis. They are THE reference on antenna design/theory. 

This is a very good question and is key to fundamental EE principles. 

To help visualize this, let's picture a quarter wave center-fed dipole antenna, which resembles a 'T' shape. When an incident electromagnetic wave of the same frequency at which the antenna is resonant, there is a standing wave generated at either end of the "T". This is generated because the charges generated by the incident wave on the antenna are not continuous along the antenna. So in theory you could take a voltmeter with both probes touching the same end of the antenna and gradually move one probe to the other end, and you will see a difference in voltage. 

If you take a look at the book Electromagnetics Explained, it will show this distribution and explain (fairly well) the linkage between Maxwell's Equations and a very basic but accurate idiom on why the charges behave the way they do. 

I know this is a somewhat vague answer, but a simple forum post like this doesn't do justice to really explain the interaction of EM waves and their behavior.