Carboxylic acid strength due to the inductive effect
The strength of the carboxylic acid (and any other Bronsted acid, for that matter), is related to the 'stability' of its conjugate base, the carboxylate anion. The carboxylic acid and the carboxylate anion are in equilibrium with one another, and the relative acidity of carboxylic acids depends upon the position of this equilibrium. 

One of the main reasons why carboxylic acids are acidic is due to the ability of the charge to be delocalised around the pi-system:



However, in addition to this delocalisation about the pi system, extra stability can be gained by the presence of electron withdrawing groups adjacent to the carboxylate group (i.e. -I groups attached to the R group in the diagram above). 

These electron withdrawing groups draw electron density away from the carboxylate anion via the sigma system (the single bonds). This is advantageous as the stability increases with the number of atoms it can be spread across (i.e. how diffuse the charge is). 

Electron releasing groups (+I as you call them) destabilise, by increasing the electron density around the carboxylate group, which of-course makes it less favourable for the carboxylic acid to lose a proton and hence makes it less acidic. 

The +I groups aren't involved in breaking the O-H bond to any great extent, as you allude to. This is because when the proton leaves, it transfers its electron density to the oxygen it was attached to, cleaving the bond and allowing it to leave as H+. If the +I groups were donating into this bond, you would have to have the H leaving as H-, which is rarely observed.