``Behind The Scenes``
Let n=3 Members={a1,a2,a3,b1,b2,b3}
Total 6 players.
If the Teams were labeled Then things we go like this
A 
B 
C 
(a1,a2) 
(b1,b2) 
(a3,b3) 
(a1,a2) 
(a3,b3) 
(b1,b2) 
...... 
...... 
....... 






What above table says that in 3 teams A, B, C we put our team players in those pairs i.e. (a1,a2)(b1,b2)(a3,b3).
Then among A, B, C can arrange these pairs in 3!=6 ways. Correct? that is why 6 rows in the table.
But as soon as I'll remove the label A, B, C from them, all these 6 arrangements become 1. i.e. means we need to divide by 6 the total no. ways to arrange 6 members into 3 teams.
Similarly, when we remove labels from nteams in which 2n members were arranged, we need to divide by n!.