In Spanish, when we use the verb "wake up" or "despertar", we can use it in two ways. The first is similar to English, where we use it as a normal verb, and say something like "And then, I woke up." which would be translated to "Y luego, desperté." But in Spanish, for God knows what reason, we also use "wake up" in a reflexive form, so you can say "Hoy me desperté temprano". So you add words like "me", "te", "se", "nos", etc. for the different persons, singular and plural, so in other words sometimes we use "despertar" in a similar way to verbs like "wash" or "lavar". "El se lavó los dientes." (He brushed his teeth.) It's a bit confusing because in English you don't add a "myself" or a "yourself" to certain verbs as we do in Spanish. So that's it, the word "se" in that phrase you asked is the spanish reflexive pronoun for the third person plural "they" (my eyes). You could also just say "que mis ojos despierten" and the meaning would remain the same.