Feeding and Training are the two ways to level up your Magikarp. Training is dealt with on the previous page.
When your Magikarp goes back home after battles in the league for instance he arrives at his pond. There is usually food there, to grab it just swipe across each piece of food.
Like Training tasks, food can also be levelled up so it gives more Jumping Points (JP) for each bite. You can also unlock more different types of food which are more effective at giving JP as you play through the game and earn more coins.
As we said on the Training page, you should really level up your Magikarp a lot before taking on a league. Try not to lose without you Magikarp being maxed out. Raise it to it's full potential!
Watch out for certain Pokemon in your pond which you can tap to start a Food Frenzy and get a ton of food to increase the JP of your Magikarp.