Making a list that I’ll update from time to time so that I don’t lose track of these interviews. Happy to take more suggestions!
(1) Julie Rehmeyer on Wild about Math:
Julie Rehmeyer’s “Inspired by Math” interview
What’s always stuck with me from this interview is the story that begins around 31:30 and in particular the part beginning around 34:40 about proving that 0 + 0 = 0. It is a beautiful lesson about learning math.
(2) Nalini Joshi on the Australian TV show “The Weekly”
This is a stunning interview.
(3) Ken Ribet interviewd by Numberphile
Anyone wondering what research math is like – and especially anyone looking to go to graduate school in math so watch this video:
(4) The story about Paul Erdos on Relatively Prime
Moving in a different way that the Nalini Joshi interview, but I was captivated for the entire hour of this interview. Ron Graham’s stories are just incredible:
Relatively Prime’s story about Paul Erdos
(5) Ed Frenkel interviewed by Numberphile
(6) Steven Strogatz interviewed by Kara Miller
Mike's Math Page 
There’s almost too many to mention, but I’ll toss in Krista Tippett’s podcast interview of Keith Devlin for her “On Being” radio show (2013):
http://tinyurl.com/k54fvfu
(stop me before I remember too many!), but these “Science Lives” interview-bits (short but wonderful) from the Simons Foundation, include several great mathematicians, and probably aren’t familiar to many readers:
https://www.simonsfoundation.org/category/multimedia/science-lives/alphabetical-listing/