"It doesn't take a lot of strength to hang on. It takes a lot of strength to let go." - Henry Ford
Think of yourself as anchored in some deep part of the ocean; the weight keeps you held in place or fixated on an event, person, failure, dream, idea, etc. You are chained to this anchor, a crucial part of your mind, and everything revolves around that anchor as you try to navigate the ocean. In order to move forward in life, it is impossible to simply "let go" of that weight and let yourself flounder away with the direction of the waves. You'll have to gain the strength to finally haul up that anchor and ground yourself elsewhere. This is the hardest part: to experience the full force of that weight on yourself so that you can move forward.
For example, you may have experienced a breakup recently and it's hard to get that person off your mind. You think of them all the time: what you could have done differently, the good and bad memories, whether you should contact them, how you can stay in their life, etc. You can try to ignore them and simply "let go," whatever that really means to you, but you're still weighed down and revolving around them. Even if you meet new people it can seem impossible to reach them because you're still stuck in place, that other person is still at the back of your mind.
You'll have to allow yourself to experience the pain, the disappointment, the anger, the confusion - all of the difficult emotions - to become stronger than the weight that tries to keep you down. There has to be a better place you want to arrive at to truly motivate yourself to take the right steps to get there. It's hard to leave the hell you know for whatever other hell may lie in those unknown waters, we want to be able to have some inkling of a better place.
If you want to "let go," you have to "hold onto" the new or the hopeful. You may want to be better at your field of expertise, you may want to focus on deeper connections with people, you may want to go to graduate school, declutter your life, experience more fun, get a new job, etc. Whatever it is, you have to have some idea of where you want to go. Though specific plans for the future seldom turn out how you predict, they're absolutely necessary to move you forward to better and more exciting places in life.
To draw from personal experience, I had always wanted a dog--and I thought about adopting one for a couple of years before I did. After a relationship ended with someone who was important to me, I was able to move forward by fully processing what had happened and what I need in the future: the love and companionship that my golden doodle Masha was able to provide, and I am so grateful that my experiences with him have led me to this point. In a million years I would never have been able to predict that course of events, but that didn't change what I really need to feel happy and well in life. Figuring that out gave me the strength to hold onto the new and propel me into a multitude of other wonderful places where I've ended up.
"Letting go" of ___ and "moving on" from _____ are anchored in the undesirable. Re-anchor yourself in the desirable by, in turn, grabbing hold of the new.