the 1000 number might be large, but definitely in the ball park. Designing a microprocessor is not that easy. Besides the ARM core(s), you have the graphics cores, custom DSPs (I heard they have Siri-specific cores in the A5), power control logic, various IO macros and other random stuff. You need an architecture team to figure out what needs to go there and how things interact, a micro architecture team to develop the RTL, a test team to make sure the RTL behaves as planned, a physical implementation team to transform all this mess into polygons that run at desired speeds, various tool/methodology support teams to develop internal flows, evaluate tools and make sure everybody is following the rules and nobody forgot anything, plus various levels of management.
note that most probably they are working on a couple of different designs at the same time.
And that's before you start picking apart the system-on-a-chip stuff, which involves design and verification for all the components that the processor talks to, and tooling, layout, &c.
It's not inconceivable that Apple would have many teams working on many different ideas and designs that compete with each other. Apple is brilliant at understanding you need to disrupt yourself, before other companies disrupt you.
As for "how can you have 1000 people involved", well, a modern chip can have a billion transistors on it. The MOSTek 6502 design team was nine people, although maybe you could argue that only five of them were design engineers. The chip contained about 3500 transistors, about 700 per design engineer.
If a modern design team applied the same amount of attention per transistor as the 6502 team, it would need to contain 1.3 million people.