Cryptanalyze the cipertext which was encrypted using vigenère cipher

Question

How do they become the difference of 15?

In english the most frequent letters are 1) E and 2) T so they look at the most frequent letters at Place 1,4,7,... and this are 1) U and 2) F.

See green mark in the solution.

Answer 1

They simply mean that T is the 20th letter in the alphabet and E is the 5th, so they are separated by $20 - 5 = 15$. So looking for a pair of frequent letters separated by $15$ gives us a good guess as to which letters are replacing E and T. Since U is 21st and F is 6th and $21 - 6 = 15$, F and U are that pair.

Answer 2

The difference of 15 just comes from the positions of the letters in the alphabet. T ist the 20th letter, E ist the 5th letter in the alphabet. Thus the difference of T and E in this sense is 20-5=15. The same thing is true for the letters F (6th in the alphabet) and U (21st in the alphabet).

These kind of difference or offset calculations help in deciphering Caesar (or Vigener) ciphers.