One very noticeable effect it has on me is a reduction in anxiety. Lifting weights close to failure for weeks on end tends to put my brain into a state of calm, where things affect me less. When I'm living a more sedentary lifestyle, something as simple as a person not responding to a text message can send me into an anxiety spiral, where I promptly begin imagining all the things I may have done to upset them. When I'm exercising regularly, I'm much more likely to think "oh well, they must be busy", and go about my business.

I don't know the exact physical mechanism behind this effect, but I can say that the intensity of the exercise seems to be important. Before starting to lift weights, my primary forms of exercise were walking and casual cycling, and those didn't have anywhere near the same effect.