What causes the voltage drop in a realistic voltage source when the load current increases?
Real power supplies come in all sorts of varieties, so the details of the source resistance vary. Common to all power supplies is the fact that they have internal conductors. These conductors (internal wiring) will act to drop voltage with current. This wiring will create a "real" series resistance. 

Batteries have another effect. Current is produced by electrochemical reactions, and these reactions only occur so fast. In a lead-acid battery, for instance, sulfuric acid reacts with the lead/lead oxide terminals, and once the sulfuric acid has reacted new acid must take its place, and this takes time. As a result, the maximum current is limited by the geometry of the battery plates and this shows up as drop in voltage as current increases. This is modeled as a series resistor, although there is no physical unit which causes the voltage drop.

A good power supply these days will sense the output voltage, and adjust the  voltage of the internal source to compensate for internal voltage drops, and for slowly changing loads will approximate an ideal voltage source quite closely.