Learn Russian: У меня бессонница и стресс, поэтому мне не спится. - I have insomnia and stress, so I can't sleep.

The phrase мне не спится is an impersonal involuntary state, which means you have no control over the action. While я не сплю simply states the fact "I am not sleeping" (maybe you are reading or working), мне не спится means "I can't sleep" or "sleep isn't coming to me" despite your efforts.

Why does this sentence use у меня in the first half, but мне in the second half?

These two parts use different grammar structures to talk about "me". У меня uses the Genitive case for possession to say "I have" (У меня бессонница = "I have insomnia"). The second part uses the Dative case мне because involuntary states happen to you (мне не спится = "sleep doesn't come to me").

Why is the verb спится in the "it" form rather than matching a subject?

Because this is an impersonal sentence! There is no active subject (no "I", "you", or "he" doing the action). Without a subject, Russian verbs in these involuntary constructions default to the third-person singular form (the "it" form) in the present tense, combined with the reflexive -ся ending.

How can I remember the word бессонница?