For example, how can we explain the head feature of an adjective phrase?
The assumption is related to the fact that lexical categories such as nouns, adjectives, verbs, etc. are not atomic. In fact they can be deconstructed into features.
John is an NP category with these features [+human, +male], you can go futher.
Maya is a NP category with these features [+human, +female], again go further.
Features distinguish lexical items from each other. This is why we need to look at lexical categories as bundles of features. This view carries over to functional categories. Consider this pair:
(1) She likes him.
(2) He likes her.
If we wish to deconstruct the s (agreement and tense morpheme) and see what its features are, we will discover that although isomorphic, i.e., has the same phonological shape, it is an s in (1) and an s in (2), but still there is a difference.
(1) s: [+Singular, 3rd person, +Female]
(2) s: [+Singular, 3rd person, +Male]
This rationale can be extended to your question regarding adjectives.
(2) She is tall
(3) She is short
If we assume that humans are divided by either being tall or short, then tall as an adjective is [+tall], and short in (3) is [-tall].
There are other features, it depends on each grammatical context. I can also add that tall is [+V, -N, -Tense] because it falls under the rubric of verbal predicates. It assigns an external theta role to **she*. But it's not a noun nor a tense category.