"Aamir has to come to my home."
Translation:आमिर को मेरे घर आना है।
In this sentence, the को is not going with 'मेरे घर' but with 'आमिर'.
आमिर को - It is required of Aamir
मेरे घर - to my house
आना है - to come
The postposition after 'मेरे घर' which would correspond to 'to' in 'to my house' is omitted here like it is sometimes done in sentences with the verbs आना/जाना. But though it's not explicitly present, its effect can be seen by the fact by the fact that 'मेरे घर' is in the oblique case.
अपने is used when it is referring to the noun that is the subject of the sentence (or the subject of the clause it is in if the sentence contains multiple clauses).
SInce आमिर is the subject of the sentence, 'आमिर को अपने घर आना है', अपने here means 'Aamir's'. Therefore, the sentence translates to 'Aamir has to come to his own house'.
Similarly, since मैं is the subject of 'मुझे अपने घर आना है', that is what अपने would refer to and the sentence would mean 'I have to come to my (own) home'.
In 'Aamir has to come to my home', the pronoun ('my') is referring to a different noun than the subject of the sentence ('Aamir'). So, अपने can't be used and you have to use the actual pronoun मेरे.