Mostly covered in responses above but also close formations permit you to operate as a single speaking unit for domestic phases of flying. You can depart, recover and sequence 40+ jets on an exercise without 40+ people speaking on the radio and requiring sequencing from each other.
The minimum fighting unit is a pair, the biggest for a single formation will be 4 fighters. Tactically this permits you to coordinate firepower, sensors and mutually support one another.
Domestically it keeps you as a tight, single speaking unit to and from the airfield and tanker.
Leading a 4-Ship is challenging, being a stable platform during domestic phases is critical to not spit your wingmen out of formation. Tactically it involves understanding where 4 aircraft are in 3D space, what they need to achieve for the mission and by what time. This relies on constant assessment of fuel, weapons expenditure and locations of them all. The basic premise is the leader voices the tactic and focuses on the bigger picture, everyone else maintains a set position in order to execute that tactic and deconflicts from the leader. Tactical formations (normally much more widely spread) reduces workload and increases lethality for the decision maker.
Source: Do it every day.
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