"Personal well-being is internal" means that you, not anyone else is truly responsible for your well-being and happiness. Personal well-being does not come from external stimuli (although positive, uplifting stimuli is like a seasoning to the personal well-being you develop from within).
Your innermost thoughts, dreams, aspirations, ideals, and goals contribute to your well-being. These all begin from within, in your mind. Subsequently, you take actions that build on your thinking and feelings. Positive actions make for positive well-being. Negative thoughts, etc. that lead to negative or destructive actions lead to a less-than-ideal well-being.
In effect, you are personally responsible for your well-being, depending on how you approach, view and tackle life. Others may influence your well-being, but ultimately the actions you take and the help you seek are the drivers of your well-being.
When we refer to someone's personal well-being, we are generally talking about their state of mind rather than their physical health. In other words, personal well-being usually refers to a person's psychological state of being such as their emotions and self-esteem. This type of well-being is internal rather than external. We might be able to see a person's physical health but we can only guess at their psychological well-being.
This quote refers to more than just our ability to detect well-being in others. It also refers to where a person's well-being comes from. We have to look inside of ourselves to find our own state of well-being. This is something each person has within them. While events and outside influences do play a part, each person works to shape and control their well-being within themselves.