How to deal with big output values after classification layer during training?
As you stated in your question, those numbers go into a softmax function. Another name for softmax is normalized exponential function. Softmax normalizes numbers where the sum is constrained to be 1 and each value becomes the probability of categorical membership.
In the specific case of [23.4, -21254.3, 32123.4], applying the softmax transforms them to [0, 0, 1]. The numbers become either 0 or 1 because of numerical underflow.