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ともだちにあいます。
To meet friend
My understanding of the に particle is that it is used to indicate the indirect object, for time, and for location only when the verb is a "to exist" verb like います or すみます, and if the verb is an action you use で.
I would have used を in the sentence above as friend is the direct object.
The particle に is a "grammar" word, so it is not a good idea thinking of it as "meaning" any list of English words. For particular verbs, you have to learn "case-by-case" what particles it uses to hook to different types of object. (This is very like the way you just have to learn which preposition to use in French, or which case in languages like German or Russian.)
Anyway, in this particular case consider two sentences:
ともだちに 会います。 (takes に) I meet a friend. (direct object)
ともだちを 待っています。 (takes を) I am waiting for a friend. (preposition 'for')
Now imagine a Japanese friend asks you: "Why do you say 'waiting for a friend', and not 'waiting a friend'." How do you answer? Probably just "Because that's what we say"...? In which case you have the answer to your own question. And if you find that unsatisfactory, imagine the Japanese friend saying "Well, I find that unsatisfactory..." You can sure see where this goes.
Another verb you have to learn a similar pattern is のる. Riding a bicycle (direct object) is じてんしゃに のる (particle に）. Again you just have to learn it, but there are not so many of these that it is a big problem.
The particle に, rather than を, appears because the valency of the verb 会う requires it. Instead of に, the particle と is also possible.
English meet can appear with a bare object (I met him), or the preposition with (I met with him).
While most valencies are constant across languages (most transitive Japanese verbs are also transitive verbs in English), there are clear and obvious differences. For instance, Japanese 待つ 'wait' requires the particle を, but English wait requires the preposition for, rather than a bare object.
The に particle is literally in/on/to, as in つくえの 上に 本が あります。(here it is 'on') その へやに ねこが います。(here it is 'in') ともだちに 『こんにちは』と あいさつします。(here it is 'to')
And に does not necessarily come only for います/あります。
As for why を did not come in the place of に in ともだちに あいます, it's one of the rules of the language. を occurs for objects/jobs/nouns whereas に occurs for persons/adjectives.
However there can be exceptions like に being used for objects before のる。
バスを おりる - to get down from the bus (this is standard)
バスに のる - to get on the bus (this is an exception)
In the case of adjectives, なる - to become, is always preceded by the particle に。 Eg. わたしは つよいに なりました。
