Warning: SPOILERS for War for the Planet of the Apes ahead
The Planet of The Apes franchise has had a strange relationship with time and its own continuity for about as long as it's been a franchise. The original two films are set in what turns out to be Earth's far-flung future, while the third installment (Escape From The Planet of The Apes) features characters from that future traveling back to the then-present and - it turns out - setting in motion the events that will (in that timeline) bring about the Ape-ruled future they came from; setting up a classic time travel sci-fi paradox of what point in that temporal loop can actually be called the beginning of the story.
For a quick refresher: In that original version of the narrative, an intelligent ape named Caesar leads an uprising of fellow apes being used for slave labor (dramatized in the fourth film, Conquest of The Planet of The Apes); ultimately leading to a war between humans and apes that Caesar's people eventually win (Battle for The Planet of The Apes) - which sets the stage for a world where humans regress into a sub-intelligent mute race oppressed by Ape society which lasts so long that all but the most powerful leaders of said society believe the world had always been that way. This secret is revealed by good ape scientists Zira and Cornelius and a time-lost human astronaut named Taylor (Charlton Heston) in the original film, and when Earth explodes in Beneath the Planet of The Apes of those same good apes time-travel to old Earth in Escape; where they give birth to a son named Caesar who eventually grows up to become "The" Caesar of Ape Legend.
That's confusing? Rise of The Planet of The Apes (the first of the "new" prequel trilogy starring Andy Serkis) frames itself as a more "realistic" version of the scenario played-out in Conquest, with Serkis as an experimental laboratory ape named Caesar who gains human-like intelligence from a retrovirus created by a scientist as a cure for Alzheimers. Upon encountering abuse by humans (of himself and of other apes), this "new" Caesar releases more of the virus to grant his ape friends similar intelligence and leads an armed escape into the wilderness of the Pacific Northwest. However, the bigger story is not so cut and dry: It turns out the intelligence-granting virus manifests as a devastating rapid-spread plague among humans, setting a future at least similar to the original film in motion... oh, and if you watch carefully Rise also features a news broadcast noting the launch of a spacecraft called The Icarus - the very same ship that Heston's Taylor blasted off in prior to the original.
In other words, the "new" Apes trilogy is setting itself up as either a new and/or alternate-timeline backstory for the original film or, at least, a hypothetical future story that set in roughly the same context; with genetic engineering and viral plagues swapped-out for nuclear annihilation as the "how we got here" explanation for how our planet became The Apes' planet. But while Rise and its immediate sequel Dawn of The Planet of The Apes offered a handy breakdown of how the human/ape population scales tipped, the main question they left unanswered was one that the original films also didn't officially lay out: How did humans lose their intelligence and ability to speak while otherwise still resembling modern man, while the Apes had (by the time of Zira and Cornelius) evolved into more humanoid upright-walking forms? Now, War For the Planet of The Apes finally gives us our answer: The virus underwent a secondary mutation.
As War opens, Caesar is attempting to lead his ever-growing intelligent ape collective into safety, only to find them facing renewed human aggression from a maniacal demagogue called "The Colonel" (Woody Harrelson). What Caesar at first doesn't know is that, while he'd been motivated into action in part by a belief that The Colonel was planning to ally with a larger global human army, said army was actually coming to stop him (The Colonel) from carrying out a eugenicist-extermination of humans who'd manifested symptoms of a new evolution of the virus - one that robbed humans of their ability to speak and (its implied) impedes their ability to reason.
Over the course of the film, Caesar and his closest allies become the (at first reluctant) protectors of a human child who has already been afflicted by the mutated virus; ultimately bringing her with them to sanctuary when the apes escape during the final battle sequence where (through a series of cataclysmic coincidences) both The Colonel's soldiers and the larger army coming to thwart them effectively wipe each other out. She's given the name "Nova," also the name of a human woman who allies with Taylor in the original film - an implicit nod to the idea that this Nova and other survivors like her will be the ancestors of the mute "feral" humans who will eventually populate the fringes of The Planet of The Apes.
As War is pitched as the finale to its own story, it's unclear whether or not we're meant to assume that Caesar's hypothetical descendants (and Nova's) will inhabit the original Planet or a new one organized on similar lines, but short of The Icarus blasting off in Rise this becomes the clearest connection yet between these films and the originals; finally establishing a completed link for audiences to follow from one series to the other.