When you have a lucid dream you remember the dream the same way as a normal dream. At least that's how it is with me, and I haven't heard anyone saying something different about it. I assume you have remembered a normal dream at least once. Well that's the way you remember a lucid dream too. I don't know any reason why it would be different.

If you remember a thing that happened in waking life how do you remember it then? For me remembering dreams and remembering waking life events are the same. So that's how I remember it, I hope that answered your question.

Sometimes you don't remember a dream in a whole, if you know what I mean. Like there are little parts in it that you don't remember. You can compare it to a movie; there are movies who cover 2 years, but they don't show everything from those two years, they make a movie from 2 hours. The same is with (lucid) dreams; sometimes you don't remember everything and therefore it can feel like a lot longer than it really is.