The phrase "to soften the breast" means to make someone feel less upset, angry, or defensive. It refers to comforting or soothing a person emotionally. Full definition
This will not only soften the breast tissue some, it will also draw out the nipple so it is easier for baby to latch. (lansinoh.com)
This will help soften your breasts, release some of the pressure, and make it easier for your baby to latch on properly — which will prevent cracked and bleeding nipples. (mindbodygreen.com)
If your breast is hard, you may need to pump or express enough milk to soften your breast so your baby can latch on easily. (babycenter.com)