Building up your core helps with your balance and it helps protect your insides, and it keeps your back aligned properly. Without having a strong core you run the risk of having a bad or weak back, balance issues which could result in falling and also causing injury. I would bet there are other issues but since I lack the M.D. behind my name I really don't know them all.
I know my lower back used to tighten up when I ran in highschool. Now that I have a stronger core, I can run all day, it's also good to have a strong core in terms of lifting. If I'm trying to lift something really heavy, my core muscles start firing.