Yes, it is true! Having a strong Core is synonymous with having a very strong body. Look at football players nowadays: they are strong, agile, amazing when off balance and can take a hit and keep on going. They have learned over the years through trial and error that in order to have a strong base, you need to have a very strong explosive legs and have a core that is in excellent condition. Power comes from the ground up and if you generate enough of it and come in contact with another moving body, that force is going to be given right back at you. When that force is placed back on your body, your core better be in amazing shape to withstand that shock.
The same is when you are functioning in daily life, your body is going to be exposed to various forces that can make you off balance and also force you to use your strength, power and agility to create or finish a movement. That is why functional training combined with Core strengthening exercises is extremely important.