Cross-validation is generally used to find parameters or model structure to ensure it works well on new, unseen data. If you train your model using all the data A, B and C without doing the cross-validation, you risk overfitting and ending up with a model that performs well during training, but doesn't generalize to new data.
By performing the training on two of the folds and testing its performance on the third, you can optimize for performance on the new data. Instead of maximizing predictive power on the training set, you choose parameters that maximize the performance on the unseen data.
While you would use the cross-validation to find out the optimal (hyper)parameters, you would still usually use all the data you have to train the model.
One commonly used example of overfitting is choosing too high degree polynomial when fitting a curve.
Here, if you determine the correct degree for the polynomial by using 2/3 of the points for the fitting and 1/3 of the points for testing, you would see that the high degree polynomial generalizes less well than e.g. the straight line.
In real life (when dealing with prediction problems), we are most often concerned in making sure the model works with new data, rather than fitting perfectly to the training data.