How can a fugue shift temporarily into different keys?

In a fugue, temporary shifts into different keys are primarily achieved through the transposition of the subject.1

Following the initial exposition—where the subject is introduced and then typically answered in the dominant key—composers may restate the subject throughout the piece in various keys to provide variety and development. These statements are often separated by connective melodic passages known as episodes, which help transition between these different tonal areas.1

Additionally, in more complex or large-scale works, such as those by Ludwig van Beethoven, a fugue may be explicitly divided into distinct sections that incorporate formal changes of key, as well as shifts in meter and tempo, to structure the composition.2