After completing the construction of the Chef Academy, you can train your Chefs by using Chef Cards, and Chef Tools.
In Chef Academy, you will see the Chefs ready for training highlighted in green (see reference image below). Select the Chef to train and you will see the Chef Benefits screen with benefits and other requirements to train the Chef.
You can Tap on the Train button if you have the necessary cards and tools to complete the training.
If you lack the necessary tools, you can utilize the Assists as substitutes.
If no tools or assists are available, you can always use green cash to finish Chef Training.