Why exactly did SpaceX pick Kwajalein?

This article explains it.

It boils down to several things:

1. You can launch into any orbit from Kwajalein, equatorial or polar. there aren't too many friendly launch sites where you can do that.

2. They spent millions updating facilities at Vandenberg AFB only to get constantly delayed by military launches and ultimately kicked out.

There are advantages and disadvantages regarding launch operations from Kwajalein, Musk said.

First of all, there are no population centers nearby, making range safety easier and, hopefully, a little cheaper, Musk said. Also, any orbit is achievable from Kwajalein, with being close to the equator an advantage too, he said.

The logistics of transporting equipment and hardware to the island launch locale is the biggest issue the firm is now dealing with in readying the Kwajalein site for Falcon 1 operations.

A downside is high corrosion at Kwajalein, Musk pointed out. “I don’t think there’s a place in the world with more corrosion. It’s the perfect environment of right temperature, humidity and salt spray,” he said, with conditions being combated in ways similar to precautions taken with Florida-based rocket infrastructure.

Contrasted to Vandenberg, Musk said that the Kwajalein work is being done to satisfy one entity, the Environmental Protection Agency. In California, multiple federal agencies had to be engaged, along with state and county entities.

“Although, ultimately, we do satisfy all of them, it just takes a lot of time when there are so many cooks in the kitchen,” Musk explained.