"which does not bridge the space between two conductors" is really
unclear to me and an explanation of it might help me to understand the
whole concept of partial discharge better.
The space between two conductors often consists of several layers of insulation. For example, there may be the insulation on one conductor, air between the two conductors, and the insulation of the second conductor.
In the following illustration from the Wikipedia article, three layer have developed because of a gas-filled void in a single layer of insulation. There could also be a situation with two conductors separated by air. If the insulation of one conductor deteriorates at a small weak spot, the other two layers may prevent current from flowing, but they would be exposed to a higher voltage. The caption describes a small current flowing, attenuated by the capacitive voltage divider. That is leakage current that may not be immediately harmful, but is likely to cause additional deterioration leading to a complete failure of the insulation system and a short circuit.
In some situations, there may be additional modes of current attenuation. If the partial discharge is caused by a repetitive, short-time voltage spike that is riding on a lower continuous voltage source, the source impedance of the spike and the undamaged portion of the insulation may attenuate the fault current. Once the insulation has been damaged to a greater extent, the average voltage of the supply could then drive a much higher fault current.