So it seems, as do Arduino inputs unless pulled-down or pulled-up, the human body is acting like an antenna and picking up signals from the air, but scientifically, what exactly is happening?
Scientifically, electromagnetic radiation is being converted to the flow of electrons by your body, the antenna in this case. The exact same thing happens in your cell phone, both in reception (electromagnetic radiation-->flow of electrons) and transmission (flow of electrons-->electromagnetic radiation).
I'm only touching one end of the resistor (with my body closing that circuit to neither + nor GND), meaning the circuit is still floating, so what exactly is activating the transistor base to let current flow through the LED?
Your cell phone is floating too! The electrons are activating the transistor base, and they are coming from your body, and they are coming from your body because of the electromagnetic radiation that is reaching you. In a purely electron-based circuit the concept of "floating" is important because it means there is no well-defined path for electrons to follow, but this is something else, it is a circuit with both an electron flow component and an electromagnetic radiation component. Basically, 3rd/4th year stuff in an Electrical Engineering university program, not something to be explained simply
Look up "Maxwell's Laws" if you have a few weeks of free time for study.
The Gadget Shield
: accelerometer, RGB LED, IR transmit/receive, speaker, microphone, light sensor, potentiometer, pushbuttons