Softmax function result for already normalized probabilities
Summing to 1 is just one property of the softmax function. 

The softmax function takes the exponential of each value and divides it by the sum of the exponentials of all values. This tends to cluster values towards the mean, as you've seen in your example.

While the outputs of a softmax look and smell like probabilities, their values don't actually correspond to the likelihood of sampling from that distribution. They're better off thought of as the confidence of your prediction.