According to Quran, God is close to everyone, closer than their vein in the neck, however we the humans are not necessarily close to God to hear His messages! So those who can hear are chosen and introduced as prophets and for them are defined missions to send us the messages!
What is said above is not really wrong although it is not the way the issue is usually discussed. God is beyond His creations, so beyond time and space, no creation of God can ever fully understand a notion beyond the limits of creation, we cannot even fully understand ourselves! What is really important is to know who we are, not who God is! Where we are and that why are we here? Where we have come from and where we are going to be? Our main questions are concerning ourselves. These are questions that everyone may ask himself just at any rare moment that he deeply thinks of himself and care his own being, off from the routines of everyday life. These are not merely theoretical questions in the sense of pure mathematics, these can have very serious application to our everyday life as well. If we know why we are here then we can focus on the reason why we are here and plan our rest of life based on such understandings and etc. However, the adventure of finding proper and satisfactory answers to these MOST-IMPORTANT questions needs somewhat intelligence, honesty not to ignore the facts --in believing level-- in the favor of desires, strong will not to ignore the facts --in the action level-- in the favor of laziness or withstanding people's mocking and things like that, responsibility and maybe some other characteristics. Not everyone can individually find the truth and adhere to it in his own life. However, there have been also many who could accomplish this, they were all known in their own societies as wise, reasonable, trustworthy, honest, energetic, caring about the others, responsible, mild and etc. then have one day seen or heard a strange being introducing himself an angel of God stating that God has chosen him --now that he has survived-- help His other creature to survive as well. This is again a simplified version of the whole story. We have had prophets that were previously announced by another prophet that he will become a prophet, we have had prophets that were left free to choose if they want to be prophet or not, and Luqman --peace be upon him-- who was eligible for the position but didn't choose to become a prophet, since he was wise, more wise than the one who chose to become a prophet in a similar situation!
Those who ask why Allah doesn't show himself more explicitly to us, so that we can see or hear Him, scholars bring them an analogy. Break a mirror in very tiny parts, take one part and hold it in your front, can you see yourself in it? No, you may see an eye, your nose, your lips, but not the complete body. So the suggestion is for the mirror to be bigger. Here we are all mirrors for Allah, his creations, anyone who has enlarged his mirror-ness he would find Allah reflected in his being more clearly. Human can reach a place that Imam Hussain --peace be upon him-- reached when he was whispering with his God in his Arafe Dua:
اِلهی تَرَدُّدی فی الاْآثارِ یُوجِبُ بُعْدَ الْمَزارِ فَاجْمَعْنی
عَلَیْکَ بِخِدْمَةٍ تُوصِلُنی اِلَیْکَ. کَیْفَ یُسْتَدَلُّ عَلَیْکَ
بِما هُوَ فی وُجودِهِ مُفْتَقِرٌ اِلَیْکَ؟ اَیَکُونُ لِغَیْرِکَ مِنَ
الظُّهُورِ ما لَیْسَ لَکَ حَتّی یَکُونَ هُوَ الْمُظْهِرُ لَکَ، مَتی
غِبْتَ حَتّی تَحْتاجَ اِلی دَلیلٍ یَدُلُّ عَلَیْکَ وَمَتی بَعُدْتَ
حَتّی تَکُونَ الاْآثارُ هِیَ الَّتی تُوصِلُ اِلَیْکَ؟ عَمِیَتْ عَیْنٌ
لا تَراکَ عَلَیْها رَقیبا وَخَسِرَتْ صَفْقَةُ عَبْدٍ لَمْ تَجْعَلْ
لَهُ مِنْ حُبِّکَ نَصیبا
O my God, my hesitation in (following) Your traces will result in
unlikely visitation; so, (please) join me to You through a service
that takes me to You. How can You be figured out through that whose
existence relies on You? Can any thing other than You hold a (kind of)
manifestation that You lack and thus it may act as an appearance for
You? When have You ever been absent so that You may need something to
point to You? When have You ever been far-off so that traces may lead
to You? Blind be the eye that cannot see You watching it [by eyes here eyes of hearth are meant and not eyes of face of course]. Losing
is a servant’s deal that does not dedicate a share to the love for
You.
Anyway, the prophecy is the position to bring the special messages of God to the people not being aware of the messages otherwise, but this doesn't mean that seeing the angels or hearing to them or even hearing the messages of Allah Himself is limited to the prophets. There are many who were not prophets but could see and hear the angels (examples are in Quran, the Virgin Mary, the wife of Abraham, Luqmaan, peace be upon them and etc) and heard the messages of Allah, actually not every message is a rule, the rules are necessarily stated by the prophets but even you and me may have heard some inspirations and messages from God deep to our hearts, this is what we call Elhaam (inspiration maybe?). Even animals have this inspiration like Quran reports it about bees. For many of us for whom there are some darknesses in our hearth it is not always easy to make sure if an inspiration to our hearth is necessarily correct from Allah or maybe corrupted and fake from evils, but for those who have been purified there would be no doubt that the inspiration is from God and hence truth, so that we see prophets in history to whom God was talking, through His angels while they were asleep. While we cannot always rely on our dreams but the prophets' dreams are always honest dreams.
EDIT. Also let me add this that every prophet was also a good pattern for his people, so that no one can claim he was not able to live by the laws of God at his time, since there have been at least one person among them at the same time that was living by the rules of God, in more or less similar situations. So that God has always sent His rules with a guide who can interpret the rule for different situations and himself is a good pattern to learn from.
