Contaré means "I will count" or "I will tell." While "contar" is a common verb, its future tense form here is interesting because it implies a deliberate action of keeping track of time.
In the song, the line "Yo contaré a las horas" (I will count the hours) suggests a longing and anticipation, emphasizing the pain of separation and the hope for a reunion. It's a simple yet poignant way to express the passage of time in the context of love and absence.