Why is "lucked out" such a good thing to be?

It means that you are replete with luck. Think of similar phrases like 'all decked out', e.g.

The house was all decked out with balloons and banners for the birthday party.

and

Spread the sheet out

Out in all these contexts means 'the fullest extent or amount'.

In fact it means the same in 'out of luck', meaning you have absolutely no luck (no luck to the fullest extent). It's just the phrase is understood differently.

out of luck - luck has run out; is gone, depleted

lucked out - escaped/got out of a potentially bad situation, or emerged/came out ahead, thanks only to luck