"I have to go to Delhi to meet my friend."
Translation:मुझे अपने दोस्त से मिलने दिल्ली जाना है।
You use अपना, अपने etc when the possessive pronoun refers to the subject of the sentence. Eg: They are in their house (वे अपने घर में हैं), Raj loves his wife (राज अपनी पत्नी से प्यार करता है), I am eating my mango (मैं अपना आम खा रहा हूँ). You would use मेरे when the subject of the sentence is not मैं .For example, Neha is in my house (नेहा मेरे घर में है)
This is an example of the 'verb infinitive+है' construction that is used for compulsions ('have to') or desires ('want to'). In this construction, the subject should be accompanied by the postposition को (if noun) or be in the dative case (if pronoun). Pronouns in the dative case are मुझे, तुम्हें, आपको, उसे etc.
Eg: नेहा को मिठाई खाना है - Neha wants to eat the sweet
आपको अब दफ्तर जाना है - You have to go to the office now
Since the subject is in the oblique case in this construction, है doesn't need to be conjugated to it and it remains as है (or था, होगा in the past and future tenses).
Note that this construction is somewhat idiomatic and its meaning is context-dependent. If you want to be clear as to whether you're talking about a desire or a compulsion, it's preferable to use चाहना/चाहिए instead.
It is because of जाना. Although the postposition is not written it means to go to something. (Same thing with aana - to come from) These missing postpositions are sometimes referred to as "ghostpositions", and so far I only noticed them with jaana and aana. I don't know if this occurs with other verbs, also.
So: I go to this house -> Main is ghar jaata hoon