It's obviously easier to pronounce and, perhaps even acquire, a sound or sequence not present in one's native language if one watches carefully a speaker's mouth. What is this phenomena called? Where can I read more about it?
The root phenomenon is known as the motor theory of speech perception.
(In computing the technique is called audio-visual speech recognition.)
As it pertains to learning:
Pronunciation is the result of muscle movements. Different pronunciations require different muscle movements, just like any other physical activities. The muscles physically develop with practice, as does the brain's ability to coordinate them. As to why seeing it done properly can help, it is not so different than learning to kick a ball with a certain technique.
I'm only a tyro at linguistics, but I'm reading Linguistics For Dummies (1 ed, 2012; by Déchaine, Burton, Vatikiotis-Bateson) which seems helpful. Chapters 14 and 15 especially should aid you to explore further, but are too long to quote entirely; so I quote a few pages:
[p 253, from Chapter 14:] Hearing Language Is Seeing Language
When you speak, you move — you move your face and head, your body sways, you shift your weight, you move your hands and arms. Many such movements are unconnected to speech: While talking, you can scratch your head because it itches. At other times, these same movements are connected to speech: Scratching your head emphasizes your puzzled state when you say I wonder where I put my keys? Linguists call these movements gestures. You’re generally unaware of your own gestures and may not notice that you do them even when you talk on the phone! However, when you communicate face to face, you present perceivers with linguistically relevant information from two sensory streams — sight and sound — at the same time. We show you how speech and gesture work together in audiovisual perception.
[pp 272-273, from Chapter 15:] Producing speech amodally
Signals that occur in multiple modalities — sounds from the mouth, movements of the face or hands, and vibrations you can feel — can be multimodal or amodal. Amodal signals all originate from the same activity and just happen to broadcast on different sensory channels. For example, the visible changes in lip shape and face deformations due to jaw movement result from shaping the vocal tract to produce audible speech. Researchers consider these signals to be amodal (without a specific intended modality) in terms of their production even though perception processes them through multiple sensory channels. Amodal signals don’t involve extra planning or additional neural and motor resources. Producing speech amodally means that the signals in each sensory channel have a lot of shared structure and convey the same or, at least, related information. Perceivers take advantage of this when they process speech through multiple sensory channels. For example, it’s easier to understand people face to face — when you can see them — than on the telephone. And if someone is deaf, being able to see how the movement of the lips, jaw, and even the tongue change the shape of the face can be critical for perception. This is a form of speechreading that everyone does to some extent without special training. (For more on speechreading and multisensory perception, see Chapter 14.)