If you're looking at the graph of the function, the first derivative is the slope of the tangent line. Derivatives tell you how something is changing. If the second derivative is positive, it means the first derivative is increasing.
Imagine the tangent line on a curve at a point while the point moves from left to right. If the slope is increasing, then the tangent line is rotating counter-clockwise. If the slope is decreasing, then the tangent line is rotating clockwise. So you have this rule: Second derivative positive means counter-clockwise rotation. Second derivative negative means clockwise rotation.
Now further imagine what these rotations mean about the shape of the curve. If the rotation is counter-clockwise, the curve must be concave up. If the curve is concave up, and you happen to be at a critical point, then that critical point must be a minimum. (The first derivative is $0$ here, and since the slope is increasing, it has to be negative on the left and positive on the right.)
In sum, second derivative positive means tangent line is rotating counter-clockwise. In turn, this means the curve is concave up. In turn, this means a critical point is a minimum.