If it was only "lui gira", it would mean that "he turns". Sure, in the English translaton of this sentence, we can guess that the meaning is that he is turning himself, but in Italian you need to be more precise. Lui gira? Cosa gira lui?
What this sentence is using is called a reflexive verb. A reflexive verb is one where any action performed by that person is done unto that same person. Examples of this kind of verb are lavarsi, girarsi, chiamarsi, svegliarsi, etc.
I wash my cat (Lavo il mio gatto). I wash myself (Mi lavo).
You wake up your sister (Svegli tua sorella). You wake yourself up (Ti svegli).
"He turns to her" implies he is turning himself. "girarsi" means that somebody/something is turning. "La lampada nel faro si gira" >>The light in the lighthouse is turning. You would not use "itself" here. Translating "mi giro" as "I turn myself" is too much. "He turns himself towards her" may be acceptable English but it is not the correct translation of the Italian sentence.
Actually it is a reflexive pronoun AND that is part of the reflexive verb.
http://www.locuta.com/epronrif.html Italian Reflexive Pronouns
Reflexive pronouns are similar to direct object pronouns, except in the third person singular and plural ("si").
Mi guardo allo specchio
I look at myself in the mirror
Ti guardi allo specchio
You look at yourself in the mirror
Si guarda allo specchio
She/he looks at her/himself in the mirror
Ci guardiamo allo specchio
We look at ourselves in the mirror
Vi guardate allo specchio
You look at yourselves in the mirror
Si guardano allo specchio
They look at themselves in the mirror
The website has some helpful tables as well.