To be almost finished; to be on the verge of ending. The -kakeru form indicates something is nearly complete.
会議が終わりかけるところで彼が意見を言い出した。
He started to give his opinion just as the meeting was about to finish.
ケーキが焼き終わりかけたら、オーブンの温度を下げてください。
When the cake is almost done baking, lower the oven temperature.
雨が止みかけると、道が一気に明るくなった。
When the rain was about to stop, the road suddenly brightened.