to fix something by agreement: to set a person, time, place, or task in advance, and so to come together at the arranged point
At its heart this verb is about setting something firmly ahead of time. Someone appoints a person to a role, names a time, or hands a task to an agent, and from then on that arrangement is fixed. It is the language of a promise kept, an assignment made, a rendezvous agreed.
Because an appointment is something two sides share, the verb naturally turns reciprocal. When people (or God and people) keep an arranged meeting, they are said to come together at the agreed place. This is where the noun for a sacred "appointed time" and the phrase "tent of meeting" come from. The same root that fixes the appointment also names the keeping of it.
The senses below simply follow that arc: first the act of appointing (a person, a time, or a task), then the meeting that results, which ranges from God graciously keeping his rendezvous with his people to crowds gathering, often to plot or make war. The stronger forms push it toward summoning an opponent to a showdown, and the passive toward simply being set in place.