Put yourself in the shoes of a bond investor who is about to choose between a callable bond and an option-free bond (both with the same coupon and tenor and par value).
When you get more benefits from the asset, you will pay more for it (value it higher). When you have disadvantages in holding the asset, you pay less for it (value it lower).
So in the case of the callable bond, the call option is a disadvantage to the investor, and with that uncertainty, means the investor will want a higher yield from the callable bond (relative to the option free bond). Higher yield translate to a lower price.
Value ~of ~callable ~bond = Value ~of ~option ~free~bond - Value ~of ~call ~option
But for a putable bond, the put option is a benefit to the investor, so the investor will pay a premium for it (hence value of putable bond > value of option free bond)