Engalana comes from the verb engalanar, meaning to adorn, beautify, or dress up. It's a more elegant and less common word than simply saying 'decorar' (to decorate).
In "El Día Que Me Quieras," the lyrics say, "la rosa que engalana se vestirá de fiesta" (the rose that adorns itself will dress up for a party). It paints a beautiful, almost personified image of a rose preparing itself for a special occasion, highlighting the celebratory and romantic mood of the song.