It just means that the to infinitive can modify the meaning of a noun (request) or an adjective (keen).

The reference to "non-finite clauses" makes little sense since all five of the examples are non-finite (because none contains an active verb). An infinitive, like "to sing", is the NAME of a verb. It does not act as a verb since it has no subject. Although it has no subject, it can have an object or indirect object. It commonly plays three roles: adjective, noun, or adverb. In each of these roles it can anchor a clause of the same type. For instance, as an adjective it can anchor an adjective clause. Here are some examples: 1. ADJECTIVE "This is the song to sing" ("to sing" is an adjective that modifies "song") "This is the man to sing the song" ("to sing" modifies "man" and anchors a clause in which "song" is its object.) 2. ADVERB "It is nice to listen to that song." ("to listen" modifies "nice". "That song" is an indirect object for "to listen".) 3. It can be a noun. Examples: "I like to sing." ("to sing" is the direct object) "I like to sing Bach cantatas" ("to sing Bach cantatas" is a noun clause and the direct object. "Bach cantatas" is the object of "to sing".) Your examples: "a request to see someone" ("to see" is an adjective modifying "request". "Someone" is the object of "to see") "keen to get on" ("to get on" is an adverb modifying "keen")

Well, the difference is very subtle and might appear non-existent, but if you really analyze the phrases, you might see that they are not exactly the same structurally. 1) "relate to the meaning...". It means the infinitive explains the noun or adjective. "to see someone" is appositive (restates or explains) to "request". You can't rephrase it with a relative clause such as "that wants to see someone" (nonsensical since a request can't have a mind) or "a request that you want to see someone". The latter sentence makes sense, but "that" here is not a relative pronoun but a conjunctive explaining what "request" is. If it is a relative clause, you must be able to identify the role it plays in the clause but there is none since "you want to see someone" is a full sentence missing no part. "keen to get on" is similar. "to get on" just explains what the person is keen about. 2) "form a type of non-finite relative clause". It is equivalent to a relative clause. "the man to save us" = the man who will save us. ("the man" = subject of the rel. clause) "the method to use" = the method we/they are going to use. ("the method" = object of the verb "use") "nice to listen to" = (someone/something) who/which is nice to listen to. (object of the preposition "to") "keen to get on" and "nice to listen to" especially look alike, but only the latter has a direct structural connection between the adjective and the infinitive. You can rephrase it like "It is nice to listen to THEM/IT", but you can't do that with the first sentence.