Applications of sunflower lemma in theoretical computer science In one lecture by Kewen Wu who is one of the authors of paper 


  Improved bounds for the sunflower lemma,


it is said that the sunflower lemma can be applied to many fields like


circuit lower bounds
data structure lower bounds
matrix multiplication
pseudorandomness
cryptography
property testing
fixed parameter complexity


What I can know about the sunflower lemma comes from the books 


  Parameterized Complexity Theory
  
  Extremal Combinatorics 
  
  The Mathematics of Paul Erdős II     2013


and papers 


  Improved Bound On Sets Including No Sunflower
  With Three Petals Junichiro Fukuyama 2018
  
  Improved bounds for the sunflower lemma Ryan Alweiss, Shachar Lovett, Kewen Wu, Jiapeng Zhang 2019


How is it applied to the field of computer science?
Sunflower lemma has applications in data structure lower bounds(as mentioned above). For eg. see:
Lower Bounds on Non-Adaptive Data Structures Maintaining Sets of Numbers, from Sunflowers.