I've done a lot of rock climbing, so I know firsthand the importance sometimes of not letting go! This applies to other things as well: keeping hold of a child's hand while crossing the street, staying true to your ethics in a tricky situation, or sustaining attention to your breath while meditating.
On the other hand, think of all the stuff - both physical and nonphysical - we cling to that creates problems for us and others: clutter in the home, "shoulds," rigid opinions, resentments, regrets, status, guilt, resistance to the facts on the ground, needing to be one-up with others, the past, people who are gone, bad habits, hopeless guests, unrewarding relationships, and so on.
Letting go can mean several things: releasing pain; dropping thoughts, words, and deeds that cause suffering and harm; yielding rather than breaking; surrendering to the way it is, like it or not; allowing each moment to pass away without trying to hold on to it; accepting the permanently impermanent nature of existence; and relaxing the sense of self and opening out into the wider world.
Living in this way is relaxing, decreases hassles and conflicts, reduces stress, improves mood and well-being, and grounds you in reality as it is. And it's a key element, if you like, of spiritual practice. To quote Ajahn Chah, a major Buddhist teacher who lived in Thailand:
If you let go a little, you will have a little happiness.
If you let go a lot, you will have a lot of happiness.
If you let go completely, you will be completely happy.
Appreciate the wisdom of letting go, and notice any resistance to it: perhaps it seems weak to you, foolish, or against the culture of your gender or personal background. For example, I remember talking with my friend John years ago about a woman he'd been pursuing who'd made it clear she wasn't interested, and he felt frustrated and hurt. I said maybe he should surrender and move on - to which John replied fiercely, "I don't do surrender." It took him a while to get past his belief that surrender - acceptance, letting go - meant you were wimping out. (All ended happily with us getting drunk together and him throwing up on my shoe - which I then had to surrender to!) It takes strength to let go, and fortitude, character, and insight. When you let go, you're like a supple and resilient willow tree that bends before the storm, still here in the morning - rather than a stiff oak that ends up broken and toppled over.
Be aware of the letting go that happens naturally all day long such as, releasing objects from your hands, hanging up the phone, pushing send on an e-mail, moving from one thought or feeling to another in your mind, saying bye to a friend, shifting plans, using the bathroom, changing a TV channel, or emptying the trash. Notice that letting go is all right, that you keep on going, that it's necessary and beneficial. Become more comfortable with letting go.
Consciously let go of tension in your body. Exhale long and slowly, activating the relaxing parasympathetic nervous system. Let go of holding in your belly, shoulders, jaws, and eyes.
Clear out possessions you don't use or need. Let in how great it feels to finally have some room in your closet, drawers, or garage.
Pick a dumb idea you've held on to way too long - one for me would be that I have to do things perfectly or there'll be a disaster. Practice dropping this idea and replacing it with better ones (like for me: "Nobody is perfect and that's okay").
Pick a grievance, grudge, or resentment - and resolve to move on. This does not necessarily mean letting other people off the moral hook, just that you are letting yourself off the hot plate of staying upset about whatever happened. If feelings such as hurt still come up about the issue, be aware of them, be kind to yourself about them, and then gently encourage them out the door.
Letting go of painful emotions is a big subject, with lots of resources for you in books such as Focusing, by Eugene Gendlin, or What We May Be, by Piero Ferrucci. Here's a summary of methods I like: relax your body; imagine that the feelings are flowing out of you like water; vent in a letter you'll never send, or out loud someplace appropriate; get things off your chest with a good friend; take in positive feelings to soothe and gradually replace the painful ones.
In general, let things be pleasant without grasping after them; let things be unpleasant without resisting them; let things be neutral without prodding them to get pleasant. Letting go undoes the craving and clinging that lead to suffering and harm.
Let go of who you used to be. Let yourself learn, grow, and therefore change.
Let go of each moment as it disappears beneath your feet. It's gone as soon as you're aware of it, like a snowflake melting as soon as you see its shape. You can afford to abide as letting go because of the miracle - which no scientist fully understands - that the next moment continually emerges as the previous one vanishes, all within the infinitely tiny duration of Now.