What does it mean to live in the present moment and let go of your past while being open to a future that may or may not happen?

Living in the present moment means letting go of the past without any condition for the future. It means living your life consciously i.e. being aware of everything that is going on with your thoughts, the words spoken, the actions of the body, and appreciating the environment around you.
The ability to be in the present moment is a major component of mental wellness and enjoying the here and now. Past and future exist in the mind whilst living in the present moment are in the now.

The way to let go of the past or to stop yourself from thinking about the future is to understand the science related to revisiting the past and dwelling in the future and also its implications.

Suffering from stress, worries, uncertainties, etc, arises when one starts to think about the past or worries about the future. Furthermore, when one worries about the future, one only succeeds in ruining the outcome.

Instead of living in the present, we live for the most part in our minds, cluttered with thoughts. These thoughts are either related to our memories from the past, nice or horrible, or our imaginations regarding the future, again could be nice or horrible. If they are nice, we react with deep attachment towards them and if horrible, in reaction we detest or hate them. For example, all year round, you look forward to your holidays, and when you finally get there instead of enjoying the holiday, you spend most of your time thinking about your work. Hence, you cannot fully relax and appreciate the environment around you because your mind keeps going back to the workplace.

These memories prevent us from being in the moment. As a rule, the greater the like or dislike towards something, the more consumed you will be by the memory of it, and thereby you are living in the past or thinking about how you can change the future related to it. These memories cannot be altered and there is nothing to gain by living in them.
What has happened is the past and the future is not in our control. The only thing useful is the present moment. Hence, we should live in the moment, in the present and enjoy all that is in front of us.

Living in the present can only happen after acquiring Self-realisation. Through attaining the knowledge of ‘who am I’, you acquire the right understanding that ‘I am not this body and mind, rather I am a pure Soul.’ What is the nature of the Soul? It is to see, know and constantly remain in a blissful state.

So after Self-realisation, as you begin to learn in the awareness of the Soul from moment to moment, you naturally stop living in the past or worry about the future. And even when you experience sadness when surrounded by memories, or you experience tension and worry for some future event in life, you only see it and know it by maintaining the awareness of the Self. This is how we can remain in the present moment, always.