Electricity only flows in a complete circuit, so how does TDR (time domain reflectometery) work?
The statement that electricity flows only in complete circuits is a simplification most suitable for so-called lumped systems, where all of the elements (e.g. capacitors, resistors, etc) are connected by short and relatively ideal wires, and the wavelength of any waves are much longer than the physical dimensions of the circuit. For example, the wavelength of a 100 kHz radio wave is on the order of single-digit kilometers, so a lumped-circuit model is suitable for discussion the operation of e.g. a linear audio amplifier.
Lumped system modeling is not an adequate model for systems where your signals have wavelengths shorter than the circuit elements themselves -- in that situation, distributed models and electromagnetic theory are better descriptions and the lumped-circuit model falls apart. This theory is commonly seen in microwave and high-speed radio circuits, where even the shape and positioning of wires is key to achieving the necessary performance goals.
In the distributed-element model, a transmission line can be modeled as a medium where voltage and current waves travel under particular constraints. The key ones are: the propagation speed (how fast a disturbance moves down the line), characteristic impedance (the ratio of voltage to current waves in a disturbance traveling on the line), and loss tangent (how much the disturbance decays as it travels).
Under these assumptions, discontinuities (where the characteristic impedance of the line changes) must lead to a reflection as a result of mathematical boundary conditions at the discontinuity. Time-domain reflectometry relies upon this exact mechanism, transmitting a sharp pulse and noting when reflections return to the source. This is not unlike ultrasonic inspection or sonar detecting cracks or objects in an acoustic medium.