There are a couple of things you need to do, prior to correctly using the EarlyStoppingCallback()

from transformers import EarlyStoppingCallback
...
...
# Defining the TrainingArguments() arguments
args = TrainingArguments(
   f"training_with_callbacks",
   evaluation_strategy ='steps',
   eval_steps = 50, # Evaluation and Save happens every 50 steps
   save_total_limit = 5, # Only last 5 models are saved. Older ones are deleted.
   learning_rate=2e-5,
   per_device_train_batch_size=batch_size,
   per_device_eval_batch_size=batch_size,
   num_train_epochs=5,
   weight_decay=0.01,
   push_to_hub=False,
   metric_for_best_model = 'f1',
   load_best_model_at_end=True)

You need to:

1. Use load_best_model_at_end = True (EarlyStoppingCallback() requires this to be True).
2. evaluation_strategy = 'steps' instead of 'epoch'.
3. eval_steps = 50 (evaluate the metrics after N steps).
4. metric_for_best_model = 'f1',

In your Trainer():

trainer = Trainer(
    model,
    args,
    ...
    compute_metrics=compute_metrics,
    callbacks = [EarlyStoppingCallback(early_stopping_patience=3)]
)

Of course, when you use compute_metrics, for example it can be a function like:

def compute_metrics(p):    
    pred, labels = p
    pred = np.argmax(pred, axis=1)
    accuracy = accuracy_score(y_true=labels, y_pred=pred)
    recall = recall_score(y_true=labels, y_pred=pred)
    precision = precision_score(y_true=labels, y_pred=pred)
    f1 = f1_score(y_true=labels, y_pred=pred)    
return {"accuracy": accuracy, "precision": precision, "recall": recall, "f1": f1}

The return of the compute_metrics() should be a dictionary and you can access whatever metric you want/compute inside the function and return.