It all depends on your perspective. In order to break chemical bonds, energy must be transferred from the surroundings to the molecule whose bond will be broken. You could say that in order to break a bond, a molecule absorbs energy from the surroundings, but it is equally correct that the surroundings lose energy due to the breakage of a chemical bond.
Similarly, when molecules form a bond, they transfer some of their chemical energy to the surroundings. Thus, in the formation of a bond, the molecule loses energy to the surroundings, which is equivalent to saying that the surroundings absorb the chemical energy released by the molecules forming a bond.
Personally, I like the organic chemists' way of looking at things from the point of view of the molecule. When in doubt, I would trust organic chemists' explanations over biologists' explanations. For example, most biochemistry text books do not have a good explanation of why ATP hydrolysis releases energy. Most biochemistry text books say something to the effect that the phosphodiester bonds in ATP are high energy bonds, so breaking the bond releases a lot of energy. A more correct explanation is that, because the phosphodiester bond of ATP is so high energy, you don't have to put a lot of energy into the system to break the bond (creating a higher energy state than ATP). Thus, when you form a new bond with water, much more of the energy released by the formation of the bond is available to be used to perform chemical work. As an analogy, it's like increasing your net profit by keeping your income the same but decreasing your expenditures.
(note: in reality the process of bond breakage and bond formation occurs simultaneously. I treat them as separate steps here in order to make the picture a little clearer. This is justified because the thermodynamics of a process is independent of the path taken. Thus, the sittuation where you sequentially break a bond then form another bond has the same thermodynamics as the case where the bonds are broken and formed simultaneously).