The answer is
Sunday
The best way to visualize it is by creating a table with values. From my point of view the correct answer is Sunday. It is the only day that never repeats.:
$\begin{array}{c|c|c|c|c|c|c|c}\underset{(Statement~\#)}{\text{Speaker}}&\text{Mon}&\text{Tue}&\text{Wed}&\text{Thu}&\text{ Fri }&\,\text{Sat}\,&\text{Sun}\\\hline1&\text{X}\\\hline2&&&\text{X}\\\hline3&&\text{X}\\\hline4&&&&\text{X}&\text{X}&\text{X}&\color{red}{\text{X}}\\\hline5&&&&&\text{X}\\\hline6&&&\text{X}\\\hline7&\text{X}&\text{X}&\text{X}&\text{X}&\text{X}&\text{X}\end{array}$
Filling in the rows of the table:
Statement 1 is true only if today is Monday.
Statement 2 is true only if today is Wednesday.
Statement 3 is true only if today is Tuesday.
Statement 4 is true only if today is in the range from Thursday to Sunday.
Statement 5 is true only if today is Friday.
Statement 6 is true only if today is Wednesday.
Statement 7 says that yesterday was not Saturday. Then yesterday could be Monday, Tuesday, Wednesday, Thursday, Friday or Sunday. So today is Tuesday, Wednesday, Thursday, Friday, Saturday or Monday – any day except Sunday.
Finally, reading down the columns of the table:
On Monday, statements 1 and 7 are true.
On Tuesday, statements 3 and 7 are true.
On Wednesday, statements 2, 6, and 7 are true.
On Thursday, statements 4 and 7 are true.
On Friday, statements 4, 5, and 7 are true.
On Saturday, statements 4 and 7 are true.
On Sunday, only statement 4 is true.
The only day when only one statement is true is the correct day. That is Sunday.
