The sensory device is a weight on a spring. It is indeed "small-scale motion of some tiny components of the sensor", and it's also "another chip on a circuit board".
The key word here is MEMS. It's possible to build small silicon structures and then etch away underneath them, leaving a free-floating piece. If the piece is long and thin, it will deform under gravity (or any acceleration) by an amount proportional to its Young's modulus. The change in position affects the capacitance between the moving part and stationary parts around it, which can be measured electronically.
Generally they have only one three-axis accelerometer. Better precision can be achieved by adding gyros or another accelerometer separated by a distance; Nintendo did this with Wiimote add-ons.
Many phones also contain a magnetometer, which tells you vaguely where magnetic north is relative to the phone, although the calibration tends to be bad on these.
Addressing specific parts of the question:
- What makes smartphones tilt-sensitive?
MEMS accelerometers. Few mm square chip package, $0.50 or less in quantity.
- Will they retain this ability in zero-gravity conditions?
Not exactly. They no longer have a convenient reference vector. However, they can still detect acceleration, so if you have one of those "lightsaber" apps and wave it around it will still work on the ISS. But neither you nor the phone have a clear idea of "up".
(The Raspberry Pi kit sent up there has an accelerometer and a bunch of programs written by schoolchildren, so there's almost certainly a video demonstrating this somewhere)
The raw output of a 3-axis accelerometer is a vector of 3 values measured in m/s^2. The magnitude of this vector will usually be about 1g, but the direction varies. For a stationary phone it will point downwards. If you move it then the acceleration vector will change direction. If you drop the phone, i.e. it goes into freefall the same as a phone on an orbiting craft would be, then the magnitude goes to near-zero. This makes the direction of the vector swing wildly and turn to noise.
The use of accelerometers as drop detectors for hard disk safety was popularised about a decade ago by Macbooks. People found other uses for them.
- how does it work?
Answered in more detail by other answers.