7 As for you, be fruitful and increase in number; multiply on the earth and increase upon it."
8 Then God said to Noah and to his sons with him: 9 "I now establish my covenant with you and with your descendants after you 10 and with every living creature that was with you—the birds, the livestock and all the wild animals, all those that came out of the ark with you—every living creature on earth. 11 I establish my covenant with you: Never again will all life be cut off by the waters of a flood; never again will there be a flood to destroy the earth."
12 And God said, "This is the sign of the covenant I am making between me and you and every living creature with you, a covenant for all generations to come: 13 I have set my rainbow in the clouds, and it will be the sign of the covenant between me and the earth. 14 Whenever I bring clouds over the earth and the rainbow appears in the clouds, 15 I will remember my covenant between me and you and all living creatures of every kind. Never again will the waters become a flood to destroy all life. 16 Whenever the rainbow appears in the clouds, I will see it and remember the everlasting covenant between God and all living creatures of every kind on the earth."
17 So God said to Noah, "This is the sign of the covenant I have established between me and all life on the earth."
Regarding the first little bit, "be fruitful and multiply," note that it is redundant -- God already said this a few verses back. In any case, it doesn't seem to mean anything very profound in context. Remember we're pretending there are only eight people in the world right now. (Yeah, unlike Cain and Abel, Noah's three sons do have wives and therefore don't need a separate act of creation to carry out the instruction. But incest is required in the second generation, a little problem Genesis just ignores.) So obviously they need to build up the population. It doesn't say to do it without limit, but people read into the Bible whatever they want to read, so some Catholic theologians have argued that this condemns contraception, and other people have said it means we should try to make the human population as large as possible. (What's the difference, the Rapture is coming anyway.) Sorry folks, I really don't see it.
As for the stuff about the covenant is also redundant (viz Genesis 8:21-22), which tells me that we are getting two versions of the same story here. I've already commented on that, but what is new here is the rainbow. The rainbow must have appeared as wondrous to ancient people as it does to us. A few weeks ago I was at my aunt's house on the north shore of the Long Island Sound, and we saw a spectacular full double rainbow that touched the water on both ends. It was impressive enough that I can certainly see why people thought it must be a message from God, and a warm and friendly one at that.
To me, it's no less wonderful a sight because I know why it happens: that light has wavelike properties, that the colors we see correspond to different wavelengths of light, that white light from the sun is composed of a range of wavelengths, that light slows down when it passes through water droplets in the atmosphere, and in the process the colors within sunlight spread apart and we perceive the rainbow. Isaac Newton figured that out in his laboratory, by using his reason to design experiments and deduce explanations for their results.
The Bible is wrong, of course. Rainbows existed for billions of years before humans came along, existed throughout our time on the earth, including before the flood, which never happened anyway, and will probably still exist long after we are gone. So they aren't in any way a message from God particularly to us. But they are an astonishing and beautiful fact of creation.