T-20 schools require a "Story"

I am applying for a PhD in Robotics targeting T-20 schools and I have discovered few references to get in touch with potential PIs. One of my references is an advisor for undergrad admissions at Stanford and he told me that while my qualifications are somewhat decent, top schools have so much competition that after a certain point qualifications do not matter and what really matters is how attractive your story and your journey is up to this point. As in he has seen students with better qualifications get rejected and lower qualifications get accepted because they had a better story to tell.

He advised me to take a week or two exclusively to work on my story and asked me to not hesitate to dramatize or exaggerate it and act like a salesperson for myself and my skills before he gets me in contact with my PIs of interest.

I want to know:
1 - How true is this?
2 - What sort of elements can I use to my advantage when crafting one for myself e.g. things that may apply to me include being an applicant from a 3rd world country, lack of domain specific resources, independent research experience e.t.c?