One of the requirements of obtaining a prescriptive easement is that the use of the property is "hostile" to the interests of the owner. This means that if the owner gives someone permission to use the property, a prescriptive easement cannot be established. The law of easements is complicated, if you are unsure of your easement rights, you should consult an experienced real estate lawyer.
Disclaimer: This answer is provided for informational purposes only, does not constitute legal advice, and does not create an attorney-client relationship. Actual legal advice can only be provided after completing a comprehensive consultation in which all of the relevant facts are discussed and reviewed.
Welcome to reading legal opinions! The statement you quote, "Further in either prescription or adverse possession, the use or possession is presumed to be in subordination to the title of the true owner, and with his permission..." means that the court, when deciding a matter of claimed adverse possession will presume that the adverse possessor is on the property with the permission of the owner, which would mean the claim of adverse possession or prescriptive easement would fail without proof otherwise. So, that means the burden is on the adverse possessor to prove that their use of the property was NOT with the owner's prior permission, otherwise their suit to gain title to the property will fail. The language there is just a statement of the presumptions that the court begins with which puts everyone on notice as to what they need to prove in order to prevail.
For a plaintiff to prevail on a claim for the imposition of a private prescriptive easement over a portion of the defendant landowner’s property, actual, continuous, uninterrupted use by the plaintiff of the defendant's land, for twenty years or more, adverse under claim of right and either with the knowledge of the defendant landowner or so open, notorious, and visible that knowledge of the use by and adverse claim of the plaintiff is imputed to the defendant. Furthermore, the use or possession is presumed to be in subordination to the title of the defendant landowner, and with his permission and the burden is on the plaintiff to prove that the use or possession is adverse. In Florida there is no presumption that adverse possession once shown to exist continues to do so. The plaintiff must, by clear, definite and accurate proof show that the possession continued for the full period required by law. Also, the limits, location, and extent of the occupation must be definitely and clearly established by affirmative proof, and cannot be established or extended by presumption and the pleadings, as well as the proof, particularly where a prescriptive way is claimed, must show a reasonably certain line, by definite route and termini. Acquisition of rights by one in the lands of another, based on use, is not favored in the law and the acquisition of such rights will be restricted. Any doubts as to the creation of the right must be resolved in favor of the landowner. See Downing v. Bird, 100 So.2d 57, 64-65 (Fla. 1958).