Describes a physical inability to speak (e.g., due to disability, shock, or being tongue-tied). Can also be used figuratively for being at a loss for words.
Means 'taciturn' or 'quiet by nature', not physically unable to speak. 口がきけない implies an inability, while 無口 describes a personality trait.
Etymology
Phrase consisting of 口 (mouth) + が (subject marker) + きけない (negative potential form of きく 'to be effective/function'). Literally 'the mouth cannot function'.