Core stability is the boilerplate expression for all the muscles of the back, abdomen, buttocks and hip that help with providing a stable posture. These muscles are used daily to a greater or lesser extent. On the bike you need them to effectively transfer power to the pedals. A better trained core ensures that the posture remains stable during the pedaling movement so that all the power is transferred to the pedal stroke as effectively as possible. At high pedaling frequencies, a strong core ensures that you do not bounce off the saddle while at low pedaling frequencies (and a high torque), for example when climbing, it ensures that all power reaches the pedal. You will notice that you need these muscles a lot if you cycle uphill, as your lower back starts to complain. The same can happen during long stretches against the wind. If your goal is to climb a mountain, it makes sense to also train these muscles regularly off the bike.
Please follow this link for our suggestions for a core stability training program - https://join.cc/core-stabililty-cyclists/