Using simpler examples first :
If there are 2 girls : Then only 1 game is played (they played with each other)
If there are 3 girls :
Girl 1 plays with Girl 2, Girl 3 (2 games)
Girl 2 plays with Girl 3 (1 game)
Total = 1 + 2 = 3
If there are 4 girls :
Girl 1 plays with Girl 2, Girl 3, Girl 4 (3 games)
Girl 2 plays with Girl 3, Girl 4 (2 games)
Girl 3 plays with Girl 4 (1 game)
Total = 1 + 2 + 3 = 6
If there are 5 girls :
Girl 1 plays with Girl 2, Girl 3, Girl 4, Girl 5 (4 games)
Girl 2 plays with Girl 3, Girl 4, Girl 5 (3 games)
Girl 3 plays with Girl 4, Girl 5 (2 games)
Girl 4 plays with Girl 5 (1 game)
Total = 1 + 2 + 3 + 4 = 10
Notice the pattern : the last number in the addition is always 1 smaller than the number of girls. As the number of girls increases by 1, one more number is added and this number is 1 bigger than the previous number in the addition.
So, 55 games = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
So number of girls = 11