fo = open('primes.txt', 'w') #tells python to open the file and delete everything in it
perhaps you want
fo = open('primes.txt', 'a') # tells python to append to the file
really you should not be doing this at all you should use with to safely open your file and do it once only outside of the loop
with open("primes.txt","w") as fo:
for n in range (x,y):
if all(n%i!=0 for i in range (2,n)):
a=[]
a.append(n)
print (">>>Writing the values to primes.txt...")
print ("##########Calculated by my prime calculator##########", file = fo)
print ("", file = fo)
print ((a), file = fo)