What exactly is the "core" and what are you training when you train it? Your core muscles are your four abdominal muscle groups - the transversus abdominis, internal obliques, external obliques, and rectus abdominis. Back muscles, too, are included in the core group - specifically the erector spinae, longissimus thoracis, and multifidus. The importance of the core muscles is their ability to provide a "center" or focus for the physical work your body is doing. If your core is not fit other muscles will have to take over, leading to the likelihood of strains, sprains, and other injuries.
Who even knew we had a core? Plenty of people did, long ago, but in those days no one talked about a "core". For many decades football coaches, ballet instructors, and gymnastics coaches trained their athletes in vigorous and strenuous techniques that all focused on core strength. High school gym teachers knew about the core. Remember squat thrusts, jumping jacks, and push-ups? All those ancient exercises (that we used to groan and moan about) train deep core muscles. We were doing core fitness before there was "core fitness".
Why do we need core fitness today? More and more our work involves sitting down. We stare at computer screens for eight hours a day. Instead of doing physical work such as farming or building, we type on a keyboard and talk on a cell phone. The long-term result is that muscles, tendons, and ligaments lost their integrity. Tight neck muscles, tight lower back muscles, and weak abdominal muscles are the result, and these issues lead to more serious problems such as chronic headaches, cardiovascular stress, impaired digestion, and depression. We need fitness activities that start building us back up again, and the right place to start is at the center - by engaging in core fitness.
The best thing about core fitness is that you don't need any equipment. You could get a mat and a physioball, but those items are optional. Take a yoga class. Take a Pilates class. Learn a few core exercises and begin to do them several times a week. You'll soon begin to notice that you feel better, in general. You have more energy. You're sleeping better. Your mood is improving. All due to a few squats, a few planks, and a few push-ups. That's a pretty good deal.