It's extremely difficult to tell what's meant by started here (whether behind us or not), because the quote comes from a passage either imperfectly translated into English or directly produced by a nonnative speaker. For example, notice the missing article a in they threw us in large camp. Started behind us as used here is definitely not a common English idiom or turn of phrase a native speaker would use.
Examining the context around the passage, I see it describes the painful state of life of Palestinian refugees exiled from Israel across the river Jordan circa 1967, probably at the end of or just after the Six Day War. The passage is set during a period of high military tension and activity in Israel (especially vis-a-vis the country Jordan and groups like the PLO), and massive displacement of Palestinians on both banks of the river Jordan.
Given this, I speculate that started refers to some kind of military activity. Not necessarily actual combat, but perhaps intelligence gathering, surveillance or general antagonism. But it's possible that I'm reading too far into the context, and started simply means started operations, made their presence known or appeared on the scene. My initial reaction is that the parallel started constructions imply that when the PLO started doing something, the FBI started doing more or less the same thing as well.
But it's not at all certain that started is supposed to mean the same thing in both cases. When the PLO started could be intended as a single adverbial clause to indicate the time frame for the remainder of the sentence. If so, whatever the FBI was doing (we don't know what) began happening at the time the PLO 'started' (whatever that means).
Behind may refer to physical placement, especially if started refers to actual combat or other military operations. If this is the case, started behind us means that the refugees were literally caught in the middle between the PLO (who were in front) and the FBI (who were behind). It might also be that the FBI was following the refugees for some reason, meaning they were literally behind us [the refugees]. Behind may also be used in a temporal sense, meaning that the FBI started shortly after we did. This reading makes less sense to me, unless us is meant to mean the PLO, which would be a grammatical mistake (because the refugees are not in the PLO), but is certainly possible.
To sum up:
- Without speaking to the author or person who was interviewed, it's extremely unlikely we'll know exactly what was meant, because the English here is degenerate.
- It's unclear what exactly is being started, or even if the PLO and FBI are starting the same activity.
- Behind most likely means physically to the rear of here, but may instead mean at a time shortly after.
- The refugees are probably us, but us might refer to the PLO, though that would be a grammatical error.
- My best guess is that the refugees were physically caught between the PLO and FBI, who were both operating in the area and most likely at odds with each other. I am not at all confident that this is right.