There are two theories as to the "rules" of time travel. One says that if you can travel back in time, you can alter history and thereby affect the present. This implies that an infinite number of alternate timelines is possible, depending on actions done in the past. E.g. you could travel back in time, and assassinate Hitler, which would prevent World War II from happening and drastically re-write history.
However, the theory that Harry supports states that the future cannot be changed. The rationale behind this is that the future is already a resultant of all past events, plus all time travel attempts to manipulate that future. Simply put, when a future outcome is known, all attempts to change it are also already part of its history, meaning they have simply failed or even lead to that outcome. Applying this to the WWII example: you could try to assassinate Hitler, but you would obviously not succeed, otherwise WWII would not have taken place anyway; the fact alone that WWII took place indicates that any attempts to prevent it have failed.
In Sphere, the future outcome is that a manned spaceship from Earth will enter a black hole, travel back hundreds of years through time and crash into the ocean. With knowledge of these future facts, this tragedy should seem easy to prevent: if one were to pass this warning to the astronauts before they embark on their voyage, they could simply stay away from the black hole and avoid traveling back into time. However, if that were to happen, there would be no ship on the bottom of the sea, and there would be no one to warn the astronauts to avoid the black hole. This would cause them to enter it anyway. Which would bring the ship to the bottom of the sea and take us back to the beginning again. This is called a grandfather paradox (named after the fact that you can't travel back and kill your grandfather, because that would prevent your own creation).
So according to the theory, the future (being the ship traveling back in time, ending up in the sea) is fixed, and the astronauts somehow never got the warning. Whatever Harry, Beth and Norman are going to try, it is already part of that future and it somehow did not prevent anything. Harry initially thinks that the reason is that everyone from the expedition eventually died, so no one could ever pass on the warning. However, as they three survived, something else should happen that prevents the knowledge from being public. They are also burdened by another dilemma: all three have a very special ability to materialize their thoughts, which could be disastrous in the wrong (i.e. human) hands. The solution is to simply forget about everything: this removes their dangerous ability, absolves them from the impossible job of trying to explain everyone about the sphere, and fulfills the condition that the timeline remains intact.
So basically "the future" knew that in the end, they would choose to forget everything, thus solidifying the timeline; causing the spaceship to never receive "the warning," causing them (the spaceship) to enter the black hole, completing everything.