It's subjective and ultimately up to the meet directors
The answer is basically: Except for the two final meets in Zurich and Brussels (for which athletes are invited in descending order based on their total DL points this season), it's all based upon athlete reputation and up to the discretion of each meet director. It's usually the job of the athlete's agent to convince the meet directors that they are good enough to race at their meet.
Of course this opens up the possibility of the DL not being as "fair" as the world championships, but I generally think they do a good job of filling the fields. It's to the meet director's advantage to only invite the fastest / highest-ranking athletes anyways, because that's what people come to see.
This is also the reason why most DL meets have a disproportionate number of athletes from the host country, even if some of them wouldn't be "good enough" to get invited to other meets. This isn't necessarily a bad thing though -- it opens up the possibility for upsets and often times these slower athletes are dragged to big PRs by running in an elite field anyways.
If a meet director were to go totally off-base and, say, only invite athletes from the host nation, it's likely the athletes would cause a stir and it's possible that the IAAF could remove it from the DL circuit. For the motivations listed above this is extremely unlikely though.
It's also worth noting that even Zurich and Brussels were up to the discretion of meet directors up until the DL format change two years ago.