I am interested in getting a more practical/layman's understanding of the Starting Value Methods in Fit Mixtures or Fit Competing Mixtures in Life Distribution than what is in the JMP documentation. Thank you!

I believe the first option assumes the data is unimodal (despite being composed of multiple distributions), the second assumes a mode for each distribution (for the most part), and the last is something in between.
Edit: unimodal is probably a bad term to use, since this can be altered based on binning algorithm. The investigator need to indicate whether they think the data forms a single group (1), completely separate groups (2), or something in between (3).