An idiomatic expression meaning that a problem or dispute is settled or that discussions come to a conclusion. The negative form (埒があかない) often means discussions aren't getting anywhere.
何時間議論しても埒があかなかったので、一旦休憩することにした。
After hours of discussion with no conclusion, we decided to take a break.
条件が折り合って、ようやく埒があいた。
The terms were agreed upon, and finally a settlement was reached.
あの問題は複雑で、簡単には埒があかないだろう。
That issue is complicated; it probably won't be resolved easily.