Many details of syntactic constructions and of pronunciation can be described by classifying constituents, for syntax, and phones/phonemes, for phonology, by whether they have some property or other. For instance, vowels may be either nasal or not, and noun phrases may be either plural or not. A non-nasal vowel is "oral"; a non-plural noun phrase (in English) is "singular" -- there are just two possibilities for the property. Such a classification is called "privative".
N. S. Trubetzkoy, part of the so-called Prague School of analysis, developed these ideas in phonology, then another member of the Prague School, Roman Jakobson, with other linguists (Fant, Halle) proposed a version of this method of analysis called the "distinctive features", which treated such privative classifications as features with binary values, plus or minus.
Although there are theoretical ideas associated with it, using binary features is a notation, not a theory. For a vowel that is nasal, I write [+vocalic,+nasal], which just means a sound segment that is vocalic rather than non-vocalic and nasal rather than oral. Phonologists very often use this notation.
If you accompany the notation with the assumption that all phonologically significant information about sounds can be represented with such binary (privative) features, this becomes a theory. But the notation can perfectly well be used without accepting such a theoretical idea.
Gerald Gazdar used feature notation in describing his theory Generalized Phrase Structure Grammar (GPSG), but the notation can also be used atheoretically, simply as a descriptive device. If you want to describe "the man" as being a constituent which is singular rather than plural and masculine rather than feminine, for instance, you could call it [-plural,+masculine], or, if you like, [+singular,-feminine].
In GPSG, this sort of description is used notationally, because it is a convenient and familiar way of writing cross-cutting classifications, like (in the above example) the gender and number of a noun phrase. Since the feature notation is not a theory -- anyone is free to use it -- using it doesn't mean you accept the theory of GPSG.
But the reason it is convenient to use it in GPSG is that GPSG is based on context free phrase structure grammar, which allows any finite set of "non-terminal symbols" in describing grammatical categories, and there is no rule that you have to use symbols that are given names spelled alphabetically, like NP, S, Adj. Instead, if you like, you can use sets of features or feature specifications as the names of nonterminal symbols. It's simply a notation, though it is associated with GPSG the theory.
Moving a feature outside the brackets that enclose a list of features, as is done in the example you gave from Carnie, is also a GPSG convention. So instead of [NP,+masculine,-plural], you can write NP[+masculine,-plural]. I don't recall that this has any significance -- it's just an informal way of indicating that you think of one feature as more important.
I haven't seen Carnie's book, but I'd guess the use of features here is an example of pointless jargon that is used to produce the impression of an esoteric and impressively technical theory. Students who have learned their linguistics under the influence of the MIT school of generative grammar often have a problem distinguishing notation from theory.