Why? What do “dreams” and “prophecy” have in common?
In both, the conscious, rational, “human” mind is relaxed.
Thoughts come spontaneously, without critical filtering.
“The imagination is man’s creative faculty…The Divine Mind communicates with the human mind through the channel of the imagination. The great truths of which man becomes convinced, or the superlative visions which flash through his mind, are but particles of wisdom transmitted by the Divine Mind, by way of the imagination.” 
The rabbis said similarly, “…from the day the Holy Temple was destroyed, prophecy was taken from the prophets and given to madmen and children.” 
“Madmen and children” — whose thought-processes can often seem (non-clinically) to be spontaneous and impulsive; expressed (even acted out) uncritically, unimpeded by rational censoring. They’re “innocent”: revealing their thoughts and feelings without deception; without acknowledgement of or regard for consequences.
Some scientists and artists strive for this, too.
The rabbis of the Talmud could hardly have believed that dreams are prophetic, per se. They said “1/60th prophecy.” That’s a rather small percentage: “1/60th” is about 1.7%. That means that almost 98% of a dream is “dross.” (Freudians and psychics might differ as to the ratio, of course.)
Yet, they meant for us to take dreams with a degree of seriousness: at least 1.7% of what we dream has real truth to it.
What is “prophecy?”
It’s commonly understood as foretelling the future, but this isn’t strictly correct. Although foretelling forms part of the prophetic literature, more broadly, a “prophet” is G-d’s “spokesperson,” delivering a message that speaks G-d’s view of how our future is being determined by our present actions. A prophecy, then, is G-d’s “comment” on our actions and their consequences. It’s distinguished from statements of moral or ethical philosophy by coming to the prophet not as the product of logic, but as an “inspiration;” an “insight;” a “vision,” that he/she can’t resist proclaiming:
“…when the Divine Mind transmits a truth to man, be it pertaining to the laws of nature, or to man’s own conduct, or to his social organization, or to the aesthetic realm, it comes without any effort on the part of man. Man then becomes a receiver, not the seeker, of truth. He becomes a channel through which wisdom flows; he becomes an instrument through which the Divine Mind expresses a truth…When man receives this truth, there is no effort on his part in the direction of attainment; the Divine Truth appears in a flash, it overwhelms the human mind, staggers its logic and strikes all its faculties with conviction.” 
A prophet might say that he/she is speaking “G-d’s word,” but means that he/she is proclaiming G-d’s view of us, using words that G-d is choosing. At least at the moment of prophecy, the prophet’s human mind/will has become as utterly relaxed as possible, allowing the Divine Mind – the higher level of our own mind – to express itself through him/her. Just as we do in sleeping/dreaming, but moreso.
So what exactly is that 1.7% of a dream that we’re supposed to take seriously?
It could be the timing. Why are we having this dream at this specific point in our lives? Why today? Why not last week? Next week? And so on. It’s “hashgachah protis” – Divine Providence.
But it could be any way that we see the dream as relating directly to our lives. Sometimes, even the understanding of a dream needs to be “inspired.”
The Divine Mind is infinite, but can be covered over or blocked by the activity of the human mind — like a small coin placed on our eye can block out the sun, even though the sun is infinitely larger than the coin. As a result, we typically experience the Divine Mind in us only in tiny bursts – of energy, or peace, or creativity, etc. We can progress in our openness to it by meditation, contemplative prayer, etc. — i.e. by giving more attention to the Divine Mind in us, than to our own thoughts.
When in sleeping we relax the human will, the rational mind, a lot of emotional material can be expressed – sometimes very creatively. But beyond the human mind itself, our higher mind – the Divine Mind in us – can express itself as an insight, a direction, or a “comment” on our lives, when our attention isn’t consumed by our thoughts, plans, or feelings.
In dreams, we become, in a sense, a prophet — even if only to ourselves.
It might seem like an intermittent, limited spirituality. Yet, the child, the madman, the dreamer, give us a glimpse of what higher realization is. The Chazon Ish speaks of “N’vu’ah” — “Prophecy” — as the goal of spiritual growth:
“…man has the power of refining his soul [לעדן נפשו] and purifying his mind [לזכך דעתו] by…knowing his Creator [בהכרת בוראו i.e. by meditation] until a spirit from above [רוח מרומם] and a sublime feeling rest upon him. Then the Creator grants him some form of connection between himself and his Creator, and he is worthy of hearing commands from…the Almighty…This we call “prophecy.” 
“Refine his soul and purify his mind” — i.e. to become as “uncritical” as a child, a dreamer or even a madman; innocently, spontaneously allowing G-d’s expression in and through us, yet tempered with wisdom and self-mastery.
It should be no surprise, then, that an extra benefit of visualized, affirmative prayer — in which we use positive statements and mental images to allow a Divine response — can be the deepening of our conviction, based on experience, of the reality of G-d’s Presence in our lives:
“If you would know G-d, do not seek merely to prove His existence, but turn to Him with your heart; affirm your union with Him, affirm His responsiveness to prayer, pray to Him; if you actually turn to G-d … speak to Him in your heart, you will be astonished to find how close He is to you, you will feel His nearness, you will have found G-d.” 
Then, we truly become what Adam and Chavah were meant to be in Gan Eiden.