1) You will need a variable that stores which day is currently selected.
public SelectedDay:string = null;
2) Then on click, set selected day,
<div (click)="SelectedDay=day">{{day}}</div>
3) Check if selected day is the same day in loop using *ngIf
or hidden
<i class="fa fa-plus" *ngIf="SelectedDay!=day" aria-hidden="true"></i>
<i class="fa fa-minus" *ngIf="SelectedDay==day" aria-hidden="true"></i>
Your final HTML should look like this -
<div *ngFor="let day of daysInAWeek">
<div (click)="SelectedDay=day">{{day}}</div>
<div>
<i class="fa fa-plus" *ngIf="SelectedDay!=day" aria-hidden="true"></i>
<i class="fa fa-minus" *ngIf="SelectedDay==day" aria-hidden="true"></i>
</div>
<div class="details">Today is {{day}}</div>
</div>
This should work.