what is the Buddhist explanation?
You're reading the Laṅkāvatāra Sūtra.
I'd like to mention just in case you don't know already that there are different schools of Buddhism, spread over centuries and continents. They have much in common and later schools evolve from earlier schools. To the extent that they're similar, it makes sense to ask about "the" Buddhist explanation.
The Laṅkāvatāra Sūtra is apparently important to the early Zen school.
This edition by Guṇabhadra is said to be the one handed down from the founder of Chan Buddhism, Bodhidharma, to the Second Patriarch, Dazu Huike, saying:
I have here the Laṅkāvatāra in four fascicles which I now pass to you. It contains the essential teaching concerning the mind-ground of the Tathagata, by means of which you lead all sentient beings to the truth of Buddhism
Essentially that all we touch, see, taste, hear, and feel is no more real than a dream.
Beware that it's some kind of analogy or metaphor: and don't take that analogy too far.
For example if something were "like an apple", by that I might mean that it's round, sweet, portable, grows on trees, and/or etc.; but that doesn't mean it's exactly like an apple in all respects.
You've focused on one aspect of "dream": i.e. that a dream is irrational, private, irreproducible, and/or not shareable ... but that's not necessarily what the suttra intended when it mentioned "dream".
There's another translation of the sutra here (by D. T. Suzuki), which begins with an explanation/commentary which you might like to read in detail.
Part of it says,
Thus, according to Buddhist philosophy, reality must be grasped in this world and by this world, for it is that "Beyond which is also Within". The Lanka compares it to the moon in water or a flower in a mirror. It is within and yet outside, it is outside and yet within. This aspect of reality is described as "unobtainable" or "unattainable" (anupalabdha). And just because it is unobtainable in a world of particulars, the latter from the point of view of reality is like a dream, like a mirage, and so on. The subtlest relation of reality to the world is beyond description, it yields its secrets only to him who has actually realised it in himself by means of noble wisdom (aryajnana or prajna). This realisation is also a kind of knowledge though different from what is generally known by this name.
When I read that, I think it's saying that we don't understand or perceive the world very well. We perceive the world as in a dream, analogous to perceiving the moon's reflection in water:
It doesn't imply that the moon is unreal or a dream, and doesn't imply that two people won't both see a reflection of the moon.
Another impression I get from that description of "unobtainable" is, sometimes you might dream that are you are with someone, that you are somewhere, that you have something ... but then when you wake up you realize, "I do not, now in waking reality, have that person or place or thing, it was just a dream". Similarly perhaps they're saying that, in just the same way, even in waking reality we don't really "have" things, things are unattainable (but I don't know why: perhaps I guess it's because all we really have are our mind-made internal sense-impressions; or perhaps it's because the having of things is impermanent and in that way dreamlike; or perhaps because we ourselves e.g. in the sense that "I have a self" are dreams).
Some of the other answers to this question are based on the perspective of earlier schools of Buddhism.
I suspect this text (or the doctrine described in this text, or the school whose doctrine is described in this text) doesn't exactly contradict those answers but it has added concepts such as "non-dualism" which I don't think are so explicit/emphasized in earlier texts/schools.
Reading more Suzuki's introduction, it says,
In considering the theory of Mind-only, we have to be careful not to understand this term psychologically.
I don't know what that means but there is a "mind-only" school e.g. mentioned in this answer, "Why is the Yogācāra school called 'mind only'?", and elsewhere.
I was wondering how Suzuki would translate "dream" but when I get into Chapter 1 of his translation I find that his translation (or the his version of the sutra which he's translating) is entirely missing e.g. the first of the sentences which contains the word "dream" in the edition which you're reading.