"Thus says the Lord, 'By this you shall know that I am the Lord: behold, with the staff that is in my hand I will strike the water that is in the Nile, and it shall turn into blood. The fish in the Nile will die, and the river will stink and the Egyptians will not be able to drink its water'."
As far as explanations go, "magic stick" is not the most satisfying, but it's about all you get in the Bible for how the Nile turned blood red, the first of the Plagues of Egypt. Though the reasoning behind it is, of course, nonsense, stories like the seven plagues in the Bible (and other legends) are sometimes based on real, if misunderstood events, which are then given supernatural explanations by those who pass them down.
What made the Nile run red with blood?
It's plausible that the Nile did, in fact, run red as the Bible suggests. In fact, earlier another legend pre-dating the Bible also mentions the Nile running red like blood. In a tale of Sekhmet, an ancient bloodthirsty Egyptian goddess hell-bent on consuming humanity, Ra poured copious amounts of dye into beer, before pouring that beer into the Nile as a sort of god prank. Sekhmet, well and truly punk'd, drank the river believing it to be blood, fell asleep after her Nile bender. When she woke up, she had forgotten about why she was killing humanity, and returned home to Ra.
What could have caused such an event, assuming gods aren't real, or they aren't as hands-on as described above? Well, one possibility is known as the Burgundy-blood phenomenon.
First proposed by Greta Hort in the 1950s, the idea is that during the reign of Ramesses II (1303 - 1213 BC) the climate of the capital city Pi-Ramesses began to change from a wet and tropical climate to become drier and more desert-like. Hort suggested that as the Nile turned from fast-flowing into a slower, muddier river, it may have provided the perfect conditions for millions of flagellates (specifically Euglena sanguinea and Haematoccus pluvalius, according to Hort) to thrive.
These algae could account for more than just the blood-red color of the river, as their toxicity would explain the death of fish mentioned in the Bible, whilst also accounting for the smell.
Stephan Pflugmache, Biologist at the Leibniz Institute for Water Ecology and Inland Fisheries in Berlin, later suggested that Burgundy Blood algae (Planktothrix rubescens) could account for the color of the water, as well as further of the seven plagues. The algae could have forced frogs to leave the river (plague two) before dying. With these predators gone, he suggests, insects would have flourished (plagues three and four), which could have led to the disease of livestock (plague five) and boils (plague six).
Yes, one of the big plagues of the Bible could actually be... algae. It's not quite as dramatic, Moses calling down a plague of pond-scum, is it?