I know this is a broad question, but I hope someone can help me understand this. I have looked at lectures and read many articles, but I find it hard to understand them (I could understand the rest of what they said).
Basically, you need the commutator of field operators to vanish for spacelike intervals. That is $$ [\phi(x),\phi(y)]=0 $$ for spacelike $x$ and $y$. This means the field operators are uncorrelated outside the lightcone, which is causal.
Moreover, the implication of this is that the transition amplitude $$ \langle0\rvert \phi(x)\phi(y)\lvert0\rangle $$ is antisymmetric in $x$ and $y$. This means, if a particle can travel beyond the lightcone, the antiparticle travels backward in the path so that the net amplitude is zero.