Can anyone help me understand the following paragraph?
I do not know if the following can be useful to explain the involuted phrase that you cited, however all the following facts are true.


A relation must have at least a candidate key and can have more then one candidate key.
By definition, an attribute is prime if it is part of any candidate key.
A transitive dependency is a dependency that derives from two other dependencies through the “transitive” Armstrong’s axiom (if X -> Y and Y -> Z, then X -> Z), so for instance if in a relation R(A, B, C, D) we have that A -> BC and BC -> D, the dependency A -> D is a transitive dependency.
A transitive dependency can be problematic (that is can produces data anomalies) if the second dependency of the transitive rule (the “Y -> Z” dependency of the rule) has a determinant which is not a full candidate key, but (a) a subset of a candidate key, or (b) contains attributes that are not prime.


Consider the relation R(A, B, C, D E), with dependencies AB -> CDE, CD -> ABE, C -> E. In this case there are two candidate keys, AB and CD, so C, the determinant of C -> D, is a prime attribute, since it is an attribute of the candidate key CD (“it is not a nonprime attribute”).

Note that in the previous example the attribute E is transitively dependent from AB (from AB -> CDE we can derive AB -> C, and from  AB -> C and C -> E, we can derive AB -> E) and the relation is not in 3NF, neither in BCNF.