The field's center circle is used for kickoff. Let's define what a kickoff is.
A kickoff is what puts the ball in play at the start of a match, second half, extra time, or after a goal has been scored.
The only players allowed in the circle are the two players kicking the ball until the ball has been kicked. So the circle keeps the other players outside of the area.
There is an arc, the half circle on the field, which is attached to the penalty box. This arc is essential for penalty kicks. Penalty kicks are free shots taken from a penalty spot resulting from an infraction that occurred in the penalty area.
When taking a penalty kick, all other plays must be behind the penalty spot until after the kick is completed. The arc keeps the other players behind the penalty kicker and ball, at a safe 10 yards away, which is the regulation. Without the arc, we would not know how far away 10 yards is.
In either scenario, the center circle and penalty arc are used for keeping both teams at a distance from the kicker or penalty kicker.