Kampf can mean fight, struggle, may even go as far as competition.
Spiele is the plural of Spiel which in most cases means game.
Battle would be Schlacht, and involve a lot of people.
School would be Schule.
Examples of use:
Es gab einen Kampf der zwei Schüler im Schulhof -There was a fight of the two pupils in the school yard.
A.H. schrieb mein Kampf - A.H. wrote My Struggle.
Nowadays Kampfspiele would be most likely be associated with computerbased combat games.
Back in the days it was more like scouting games or field games of competing groups.