Fastidio translates to "annoyance" or "bother." It's a less common word than something like molestar (to bother) and carries a slightly stronger nuance of being a nuisance or a burden.
In the song, the singer says, "Perdona si te fastidio, pero es que es mi sentir" (Forgive me if I bother you, but it's just how I feel). This use of fastidio highlights his vulnerability and the sense that he might be an inconvenience to the person he's addressing, making it a poignant and memorable choice.
"Antes De Que Te Vayas" is a farewell song. The singer is watching someone he loves leave, and instead of holding on, he asks for one last, quiet look before she goes.
She has told him honestly that she no longer loves him, and he accepts it. He tells her to take care on her path and says he will always pray for her, even though letting go clearly hurts.
He apologizes for still saying how he feels, admitting he barely has the words for it. He wishes for one last kiss to wipe away the past, but says he won't hold her back. In the end he lets her go freely, wanting nothing more than a final glance before she's gone.