Hola Kay

the verb "lucir" (to shine) is used in different ways with different meanings. "Lucir el pelo" is an idiomatic expression that, used in negative, means there is no positive effect (despite what you do), so in this sentence from the dictation:

"...a los 5 minutos de haber limpiado, ya está todo sucio otra vez, y no me luce el pelo."

This means that despite his efforts to clean, the job he's done is for nothing, because it gets spoilt by getting dirty very quickly again.

You can see an entry here for this idiom at Spanishdict. (you will need to scroll down and see the entry for meaning "aprovechar")

Saludos