Why are phosphate-phosphate bonds weaker than phosphate-water bonds?
ATP is an unstable molecule in unbuffered water, in which it hydrolyses to ADP and phosphate. This is because the strength of the bonds between the phosphate groups in ATP is less than the strength of the hydrogen bonds (hydration bonds), between its products (ADP + phosphate), and water acid anhydrides are generally prone to hydrolysis into two acid molecules.


Ignore that description Wikipedia just gave you, it is just blatantly bad from every perspective.

Phosphorous(V) is generally a strong electrophile and will be readily attacked nucleophilicly by water molecules — think of an $\mathrm{S_N2}$ mechanism. On the other side, $\ce{ROPO3^2-}$ is a good leaving group, too.

While the exact details may or may not be like I presented here, all acid anhydrides can easily hydrolysed similarly, be it disulphonic acid ($\ce{H2S2O7}$), organic acid anhydrides ($\ce{R-CO-O-CO-R}$) etc. Technically, the $\ce{P-O-P}$ bond in ATP/ADP is no different from the bonds in $\ce{P4O10}$ which we all know to be very readily hydrolysed.