Reason For Living
Something Worth Living For
noun
You know that 生きる is "to live" and 甲斐 is "effect" or "worth." Something that has a good effect on your life and gives it worth is your reason for living. From your standpoint, it's something worth living for.
Keep in mind 生き甲斐 is mostly used as a noun, so you'll see it in phrases like 仕事が生き甲斐 (my job is my reason for living) or 生き甲斐が欲しい (I want something to live for).
The readings come from 甲斐 and 生き, so if you know those you may have a reason for living, or at least a reason for doing your kanji study. Be careful of the rendaku changing 甲斐's reading to がい!
ロッククライミングは私の生き甲斐です。
Rock climbing is what I live for.