It works by always adjusting the output current to keep the voltage drop between the Vout pin and Adj pin at 1.25V(on most linear regulators), where the set resistor(s) resides. .....
Sorry for the precisation ..... this is true only for the LM317 series, that have a Vref of 1,25V ..... and, basically, they adjust the voltage, not the current directly (the current is a derivation of a certain voltage on a certain resistence load)
More specifically, the linear driver adjust the output voltage in the way that the current through the load always produce the same voltage drop on the reference resistor ..... when the load resistance increase, it increase the voltage, and when the load resistance decrease, it decrease the voltage, so it keep the flowing current in the load / sense resistor assembly always the same .....
For the LM317 serie and similar regulators, it happens keeping the drop voltage at the sides of the sense resistor (that is electrically in serie with the load) stabilized on 1,25V, cause this is the internal reference voltage of this regulators serie ..... some others have different voltages, but the principle is always the same .....
For the "current sink" and "current source" regulators types, the principle is similar, the only difference is that you use a mosfet or power transistor for regulate the voltage, and an op-amp for drive it, driven from the voltage collected from the sense resistor ..... the advantage of this system is that the current regulation is more precise, and the dropout of the assembly is lower than the one you have with the LM serie (LM317 schematic needs at least 3,75V, better 4V, more than the voltage needed from the load, so if you use a 5V FV LD, you need at teast 9V for the assembly works correctly ..... with a good mosfet/op-amp sink circuit, you may end needing 6 or 6,5V for get the same results) ..... disadvantage is that the circuit is a bit more complex .....
Also, with LM serie circuits, the trimmer, if you want to make a variable current driver, need to hold the same current, or a discrete part of it, of the load, small trimmers just smokes away for high currents ..... with a sink or source current circuit, you can just use a 1/4W miniature trimmer without problems, is only the fixed sense resistor that hold all the current, and cause it can also be 0,1 ohm, it also need to dissipate less power (example for 1A, with a LM317, you need 1,25 ohm that need to dissipate at least 1,25W, with a sink circuit with a 0,1 ohm Rsense, the resistor dissipate 100mW) .....