Positive convexity means gains accelerate and losses decelerate - a favourable asymmetry. In options this is gamma; in bonds it is the second-order sensitivity to yield.
Why it always has a price. Convexity is valuable, so you pay for it - in options through theta, in bonds through a lower yield. Nothing gives it away free.
By Jensen's inequality, convexity means the expected value of the function exceeds the function of the expected value - which is the formal reason a convex position benefits from volatility.