There are many MANY reasons horses do not stand still for people to mount, but they all lead back to anxiousness (whether it be excitement, pain, or worry about what happens next).
Sometimes the solution can be as simple as letting the horse stand still for a minute after you get on. Horses learn routines, and if you always get on and go, then your horse will soon just try for the “go” and skip your getting on.
That is obviously a very simple answer. For people who are willing to look a little deeper into their horses’ state of mind (which for your own safety I urge you to do) you want to look for signs of stress or discomfort in your horse prior to mounting.
Before you put on a saddle, check your horse for sensitivity in their back or girth area (watch their facial expressions as well as flinching in their body). Some horses that have had a lot of unpleasant experiences with saddles flinch before you even set the saddle on their back. For those horse you have to build new positive associations with the saddle and riding once you have thoroughly checked for any residual or current physical issues.
One thing important to remember is that just like when we don’t feel good we get grumpy or overreact to things we might normally let slide, horses also become less tolerant when they are dealing with even a low level but persistent pain. If you have ever had a headache that won’t go away, you can understand how your ability to “deal” becomes limited.
So once you have checked and double checked that your horse is not in physical pain before getting on, you also need to take a look at how your horse feels about the work you have been asking for while you are riding. Perhaps you went on a trail ride with some hills the horse hadn’t been physically prepared for, or you asked for way more of ______________ than you had before. Most of us would not go run a 10k without training a few weeks for it, but we don’t often think about adequately preparing our horses for carrying us (because they are horses and we see horses carrying people in pictures all the time!).
The army, which in centuries past used horses to their physical limits, came up with the rule of not making a physically fit horse carry more than 20% of their weight. So a 1000lb horse in regular physical training would not carry more than 200lbs (rider, tack, & everything).
Working with older horses or those who have had time off, I have noticed that any weight over 15% (of the horse’s weight) causes the horse’s back and shoulders to become sore even if the ride was just walking for 20 minutes. And these horses were reluctant to approach the mounting block for days afterwards.
Besides physical over exertion, there can also be mental stress. Sometimes we ask our horses to do things or go places that push them over their fear threshold and it can take days (at least) for their endocrine levels to stabilize and their emotional state to calm down.
While you work to identify and alleviate your horse’s worry, you can practice having your horse follow you at liberty to the mounting block and reward them (do they have a place they liked scratched?) and repeat until you can scratch their withers from the mounting block. When you do finally get on, get off again without asking your horse to move (do that for a couple sessions). Then start building up your rides slowly and keep focused on what makes your horse unwilling. Like young children they can’t tell us why they do things, but if you watch their behaviors and gather information without prejudice, you can make positive changes that make your rides better for both horse and rider.