In simple terms, dedicated Offset modules usually add or subtract a voltage from a signal passing through – such as shifting a 0 to +10v signal to instead vary between -5 and +5 volts.
Why is this useful? Modular synthesizers live and breath voltages – it’s the signal that tells them what to do. However, the voltages flowing around inside your system might not be in the range you need them. For example, you often use the output of an envelope generator to open and close a voltage controlled amplifier. This requires its output to go between 0 volts (closed or off for silence) and a positive voltage, such as +5, +8, or even +10 volts (to “open up” an amplifier and let sound go through). However, many envelope generators these days have loop functions that allow them to cycle continuously, converting them into LFOs (low frequency oscillators). Quite often, you want an LFO’s output to vary between a positive and a negative voltage, to vary a parameter both above and below its initial setting. For this, you need to offset its output in a negative direction to move it from strictly positive range to one that swings on both sides of 0 volts.
Useful trivia: Many front panel controls are actually offset controls, sending a specific voltage to a function of the module. A good example is the cutoff frequency control on a filter: this is usually an internal voltage added to any incoming control voltage to determine the final cutoff. It is also very common to see a “bias” or “offset” or “initial level” control on a VCA; this adds a positive voltage to the CV coming is, so that even with nothing connected to CV jack, it will pass through a sound patched to its input.