The corresponding version of the sentence with gerunds is the following:
"Running competitively means learning everything about yourself while having the time of your life"
The use of gerunds is actually the use of verbs in -ing form to represent nouns. Such nouns can be either the subject or object of a sentence.
In this case, gerunds represent activities (running, learning), but you should remember that using gerunds is not the same than using present continuous. Present contiunous is used to talk about activities in progress, that is to say, activities that are not finished yet.