First of all you need to understand the concept of "dynamic range". (Maybe you do, I don't know) The dr is the ability of your camera to render as much darktones and highlights at the same time. Have a look about this concept on youtube.
You have to be aware that the human eye has the hability to render a better DR than most of digital cameras. On average, the human eye can render 24 stops where the average camera can render 12 stops. This is why your picture doesn't look the same as what you see with your eyes.
So, this situation is a very contrasty scene where you would like to have both very dark shadows and highlights correctly exposed. Without taking the direction of post production, it is impossible to do with your camera. the better DR your camera has, the better rendering you'll have but you won't have the result expected with using lightroom or photoshop.
Using a flash is another option but then you will need good lighting material and knowledge to lit the scene correctly (I mean so that it is not obvious that you're using an artificial lighting)
When you take this picture, usually you will want to "expose for the highlights", this means you will want to protect the highlights = not burning them like in your left version. Why ? because digital sensors are usually better at keeping informations in the dark tones rather than in the higthlights. So when you will try to recover informations in lightroom for example, if you camera is not to bad, you will be able to push (to add 1 or 2 stops or more depending on your camera) your darktones, making them brighter. Trying to remove stops of lihgt from the highlights is usually more complicated not to say impossible because when it is white, totally white you can't manipulate it to recover details in this "burned" zone.
I hope this explaination makes sense and will be usefull for you. Please note that you will find a billion of articles, videos, tutorials explaining thoses basic concepts on the internet :)