A bridging shot's name is rather apt. Much like a literal bridge, this type of shot connects to otherwise disconnected moments of a story. They can seam together a jump in time or place, or otherwise repair discontinuity as called on by the narrative. They are important for maintaining the pace of a film. Do you really want to watch someone sit on an airplane for four hours? Of course not -- you want a bridging shot that tells you how characters got from one place to another without having to actually sit through it.
There are endless ways to pull off a bridging shot, but the classic example is an animated line being drawn across a map to indicate character travel. In Raiders of the Lost Ark (1981), Indiana Jones (Harrison Ford) travels from the USA to Nepal via a nice, classically-constructed bridging shot:
Other familiar bridging shots include falling calendar pages, montages of newspaper headlines, scenes of trains chugging along tracks, and seasonal changes.
The Griswold family advent calendar house in National Lampoon's Christmas Vacation (1989) is a type of bridging shot, as each instance of the shot moves the story ahead a few days, allowing the viewer to disregard any interim days the film skips.
A bridging shot doesn't have to rely on a gimmick as in the above examples, particularly if the discontinuity is short. As Durham University writes, "a bridging shot can be used to connect two shots from the same scene by using a close-up, distant pan or different camera angle thus relating the shots via content."
Instead of a physical bridging shot, editing can be used to achieve this same result. When the effect of jumping time or place is done through editing, it is referred to as a match cut, which is the basis of continuity editing. Match cuts are a transitional technique in which two shots (outgoing and incoming) are, as FilmSite describes, "matched, or linked by visual, aural, or metaphorical parallelism or similarities; there can be audio matches, segues, and visual match-cuts of various kinds." A hypothetical example of this would be a shot zooming in on two donuts which dissolves into a shot zooming out from someone's eyes. The scene moves from one place to another using a similarly-shaped visual as its reference point.
This clip shows two match cuts from Breaking Bad (2008), one visual and one auditory, which transition from scenes with Walter (Bryan Cranston) to scenes of Skyler (Anna Gunn).
Another example from Pirates of the Caribbean: The Curse of the Black Pearl (2003):
Next, from 2001: A Space Odyssey (1968). This match cut not only shows a passage of time, but thematically connects the primitive with the futuristic -- a major note for the film.
Finally, another classic example of the cut, from Hitchcock's North by Northwest (1959):
Both bridging shots and match cuts work to move across time or location in an effortless way, stylistically allowing for the pace of the film to carry without interruption.