The phrase comes from Pastoral Guidelines for Fostering Vocations to Priestly Ministry, which suggested
"it would be good for the Diocesan Office for Vocations to propose and organize an 'invisible monastery' in which many persons, day and night, are committed to continuous prayer for priestly vocations." (17)The brochure and website include some suggested prayers. They also permit you to sign up as a participant, though you can also stay more literally invisible and anonymous.
P.S. Despite the phrase "invisible monastery", these suggestions do not appear to include anything from the Liturgy of the Hours, (see Sacrosanctum Concilium 100). You can, of course, choose that on your own.