Necho II was the King of Egypt between 610 and 595 BC, who according to a single report in the Histories of Herodotus, ordered a mission to circumnavigate Africa with the help of Phoenician sailors. According to Herodotus, after a two year journey, they returned through the Straits of Gibraltar in the third year. Since the first verified circumnavigation of Africa would only occur in 1480, the idea that it was done two millennia before is certainly intriguing, and has traditionally been met with skepticism.
Lets look at what we know about a few things
Necho II: was the King of Egypt for 15 years in the 26th Dynasty, and besides this expedition, was known for two things: first, getting Egypt involved in a war in Mesopotamia, siding with the Assyrians against the Babylonian Empire, and incidentally fighting against the Kingdom of Judah on the way. Secondly, he attempted to dig an early version of the Suez Canal, a project that did not succeed. Apparently, after his reign, many references to him were removed from records, seemingly out of embarrassment, perhaps caused by disappointment with the costs of his large projects. This is easy for me to imagine, because I have seen attempts to swiftly forget leaders who decided a war in Iraq was a good idea. But based on what is in the historical and archaeological record, Necho seems to have had a penchant for ambitious and unconventional projects.
The Phoenicians and ancient seacraft: The Phoenicians were known as great mariners who, together with their colony of Carthage, managed to spread a commercial empire across the Mediterranean. The Phoenicians had cities from The Levant, through modern Tunisia, in Iberia, and down the coast of Morocco. The area that was a "normal" part of the Phoenician sphere was already about three thousand miles in length. In addition, the Egyptians themselves had been trading and voyaging around the Horn of Africa since the middle of the Third Millennium BC. In other words, the Egyptians had been familiar with almost half of the East Coast of Africa for 2000 years at that point.
Herodotus: was a well traveled Greek man who wrote reports on distant lands, based on things he heard or observed in the eastern Mediterranean. Herodotus wanted to collect as much information as he could, and often included things that he said he found untrustworthy. He traveled sometime in the middle of the 5th Century BC, or 150 years after the voyage he reported. The passage in question, from this translation on Livius.org:
Libya is washed on all sides by the sea except where it joins Asia, as was first demonstrated, so far as our knowledge goes, by the Egyptian king Necho, who, after calling off the construction of the canal between the Nile and the Arabian gulf, sent out a fleet manned by a Phoenician crew with orders to sail west about and return to Egypt and the Mediterranean by way of the Straits of Gibraltar. The Phoenicians sailed from the Arabian gulf into the southern ocean, and every autumn put in at some convenient spot on the Libyan coast, sowed a patch of ground, and waited for next year's harvest. Then, having got in their grain, they put to sea again, and after two full years rounded the Pillars of Heracles in the course of the third, and returned to Egypt. These men made a statement which I do not myself believe, though others may, to the effect that as they sailed on a westerly course round the southern end of Libya, they had the sun on their right - to northward of them. This is how Libya was first discovered by sea.
A single passage from 2500 years ago might be insufficient evidence to judge based on, but the passage by itself has some interesting evidence. Primarily, Herodotus reports that when going around the southern end of "Libya" (modern South Africa), the sun was to the North: exactly what happens south of the equator. Although some people might have known about the geometry of the earth at this point, or extrapolated based on going southwards, it seems like a very specific detail to include. Especially since Herodotus himself, and most of those around him, would not have believed it. In other words, if they were making up a story, they would have made up something more believable. Although less specific, the time given for sailing around Africa, around two or three years, is also accurate. Also, it should be pointed out that what is omitted is just as important as what is included. If this story was being made up wholesale, the ships could have been resupplied by visiting rich foreign cities, hunting the meat of dragons, or another dramatic mean. Instead, the story is relatively prosaic. Although I wonder at the logistics of sowing and reaping grain in unexplored territory, it seems believable. The exact course taken, how the tides and the winds helped or hindered the ships, is something that would take much more maritime expertise than I have to comment on.
Overall, I would say that there is no specific reason to believe that this journey happened, but there is also no specific reason to disbelieve that it happened. While the journey seems technically possible, the logistics and motivation for it are not easy to reconstruct. It might also be possible that part of the journey happened, but not all of it: that the expedition reached South Africa but then returned the same way it had came.
Other than by finding some incredible piece of new archaeological evidence, it is likely that the question of the expedition will never be settled. However, even the debate around it provides interesting ways to look at historicity. At the time of Herodotus, the people of the Mediterranean did not know that the British Isles existed, except for vague reports of an area far to the north where tin was mined. Herodotus himself only reports of Britain as a rumor. It was only 200 years later that someone from the Mediterranean world, Pythias of what is now Marseilles, would discover Britain and Ireland. Although the voyage of Pythias has its own historical puzzles, it has not been greeted with the skepticism that the report of the Necho expedition was. After the fact, ancient history was "Europeanized": the tenuous connection between the Mediterranean and Northern Europe was imagined to be stronger and earlier than historical and archaeological records show, and the evidence of Mediterranean contact with Sub-Saharan Africa and South and East Asia was downplayed or ignored. For this reason, I think that people might be prejudiced against reports of the Necho II expedition, despite it fitting in a pattern of Egyptian and Phoenician exploration of Africa.