[Taking his shirt out of the plastic bag], he shook the garment a couple of times, ran water over the bloodied neck and dried it under the hand drier, [leaving it damp, wrinkled and only a little less discoloured].
The sequence in bold is a coordination of three main clauses; they form the ‘core’ of the sentence (notice that they’re all finite clauses). By contrast, the bracketed expressions are all subjectless non-finite clauses. They don’t attach to anything in the sense that they are modifiers, though "he" is the understood subject (it was "he" who took his shirt out … and "he" who left it damp …. ).
The bracketed adjuncts are gerund-participial clauses headed by “taking” and “leaving”; they are supplements because they don’t modify anything; rather they are loosely attached expressions, set apart from the rest of the sentence by commas, and by a slight pause in speech. They provide useful information of course, but they are non-essential and can be discarded without disturbing the integrity or meaning of the core sentence.