Please explain to me why に is used instead of を in these sentences
Is that because 会います is a transitive verb? Please give me more detail.
会います (or 会う) doesn't take を because it is an intransitive verb, not a transitive verb. Transitive verbs take を, and intransitive verbs don't.
In general, the transitivity of verbs in English and Japanese is the same in, say, 80% of the time, but there are also many exceptional cases. Just because "to meet" or "to contact" is a transitive verb in English doesn't mean that the Japanese equivalent 会う or 連絡する is transitive. There are many similar examples, and you simply have to remember them one by one. Please take a look at this list: に vs. を in "to pass a test"