So, from my understanding, the bar on the side is how "wild" or spicy the horse is. I have trouble catching the ones with a higher side bar. You'll also notice the little circle icon near the bottom will also reflect the horses wildness level. The bond at the top is how much of a bond the horse has with you. You increase the bond by doing things the horse likes (either petting, giving treats, or using a lasso or whistle). You collect treats, lassos, and whistles from exploring the plains (same area you find wild horses). It takes 5% of your energy to pick up any of these items, as well as (I think) 5% energy to pet the wild horse. Doing something the horse likes will make the bond increase, and doing something they don't like will make the bond go down, or the horse run away. Once the bond gets to a certain percentage you can try to call the horse to you (a green button will appear). If it comes to you, it's yours.

Theres a lot of trial and error, and I tend to catch maybe 1 in 8 of the horses I approach.

I'm still figuring out lots of stuff related to catching wild so take this with a grain of salt (and anyone feel free to correct me)