"plays itself out" = reaches a conclusion, finishes, is complete
If something, some process is happening that interferes with another.
Such as; if someone else's priority investigation means that you are not allowed to talk to a witness, then you cannot proceed with your one untill the other is finished, has played itself out.
We cannot proceed with the reorganisation untill the labour strike is over. (has played itself out)
We cannot finish this new law untill the new government is sworn in. (the election has played itself out)