I'm assuming you are talking about the sentence 'मुझे अपने दोस्त से मिलने दिल्ली जाना है'? The subject is in the dative case there because of the 'verb infinitive+है' idiomatic construction as I've talked about on that SD.
This sentence, on the other hand, works just like its English translation. The subject is in the direct case (वह) and the verb (जा रही है) is conjugated with the subject.
It sort of works like 'meet with Aamir'.
The verb मिलना meaning 'to meet' is intransitive so it cannot take a direct object. Therefore, you cannot say something like 'मैं आमिर मिला' for 'I met Aamir'. You need to separate the verb and its object with a postposition. मैं आमिर से मिला' - 'I met (with) Aamir'.
Note: मिलना is also the verb 'to be found'/'to be given' which is transitive. In that meaning, it can take a direct object.
It's just how the language works. The person whom you are meeting is not the object of the verb but the means through which the action of 'meeting' is being carried out.
There are other verbs which work similarly and for which the connection is probably easier to make. For example, डरना (to fear) - राहुल भूतों से डरता है (Rahul is scared of ghosts). The ghosts serve as the instrument for making Rahul scared.
This sentence uses मिलने because it is in the oblique case (Verb infinitives can take the oblique case like nouns because they function as nouns). There is an implicit postposition ('ghostposition) before जा (corresponding to the 'to' in the English sentence) and मिलने is its object which is why it is in the oblique case.