Wasn't completely sure where to place the brackets but so far it's been working well. Sometimes it leans a bit forward. I have to adjust it and play with it a little so that it leans against the wall.

This is a great item and I never would have thought of a wall mount for a Bosu ball. That said I do have a few tips for you because it's a tricky job to get it mounted if you are doing it solo.
After I chose my location I just sort of placed the first mount and then held the ball and mount against the wall together with the pressure from the ball. That let the mount stay in place and I could mark around the exterior. Remove the call and then you can mark the two shorter screw mounts on the inside. For the deep ones on the back I used a screw driver to press the screws that it came with into the drywall after lining up the other holes with the marks I made previously.
That worked fine just needed to start with the screws lined up and through the holes because otherwise it can be tricky to get them into the right position to come through. Also take note of where the mounts are in the pictures. You place two at the bottom to hold the weight and one to the side above the handle which keeps it from falling forward and stabilizes it.
A key thing to key in mind is that you'll want to think about placement and having clearance to get it into and out of the mounts. The spot I picked should have been a bit lower to clear the chairs handing above since you have to come in from the side and above in order to get into the mounts. Make sure you have enough space in the direction that it needs to slide into the mounts. Right now I'm having to slide it under the chair and up then come back down so I may end up moving it lower eventually or having to relocate those chairs to get things out the way of the path of placing it in the mounts.
After marking the holes I used a 9/64 drill bit and drill the holes but sort of just wiggled the bit to make them a bit wider and I was able to get the drywall anchors in without issue and without them being too loose. The next tip I'd say is to use a nice long screw driver to get things screwed in. Also make sure and put all four screws in and get them all started before tightening it down. Its very hard to make sure its aligned and you are in the anchor and not screwing into the drywall if you get mis-aligned otherwise.
Tighten it down but not too much since you will otherwise start spinning the anchors in the wall. These won't really fully lock down into the wall at any point so you don't want to overtighten them. I placed one mount then used that as a rest for the ball while I lined up and marked the next mount. Do the bottom two first and finish with the one on the side and just about the handle last.
So that's the end of my tips, but overall I'm really happy with this purchase even if my placement leaves a bit to be desired.

This is my third attempt at employing a bracket system to hold my hemispherical ball on the wall. For me, this is the best of the bunch. I first tried something similar, but with only two brackets. That was relatively unobtrusive when the ball was unmounted, but precise placement of the brackets, just slightly below centerline, was critical to holding the ball against the wall. Lower placements were unstable, with the ball prone to fall away from the wall, assuming you could get it set in place to begin with. I don't recommend the two bracket systems.
My second attempt was to use a wall mounted metal cage. The cage is sturdy, but juts out quite a distance from the wall. Unlike the smaller brackets, you must lift the ball quite a distance up and out of the cage. That won't be an issue for many, but if your space is cramped and you have your ball tucked in behind other equipment, dismounting the ball can become awkward, or at least more effortful than is necessary.
So I have tried again, and I believe I've achieved the ultimate system for me (and likely, most others). The addition of the third bracket makes all the difference in retaining the ball in the brackets, yet it still allows for very easy mounting and dismounting. Though plastic, these are "meaty" brackets and I anticipate them being quite durable.
My single complaint with all of these brackets is the bracket mounting screws are not suitable for use directly into sheet rock. They are going to pull out of the sheet rock, and sooner more than later. The best way to use the supplied screws is to mount them to an intermediary board that is of sufficient size to allow for both ends to be secured directly into an underlying stud. The supplied screws then hold securely into the intermediary board. This issue causes me to remove one star from my overall rating–a reduction that applies to all three of the systems I've tried.
Despite the above caveat, these brackets are the ideal way to hold hemispherical ("balance") ball off the floor. I like this mounting system a lot, and my need to explore alternatives has come to an end.