Mesomeric effect? I don't see that term used too often here...are you French?
Anyways, in the US at least we tend to call the "mesomeric" effect the "resonance" effect. What it means is that the acidity increases when electrons are "pulled away" (M-) from the acidic hydrogen atom (also called a "proton") by resonance delocalization. Resonance delocalization refers to the movement of the electrons in double bonds (i.e. the pi bonds) from one atom to another by hopping from p orbital to p orbital within the double bond system. The overall effect of this electron movement is that the electrons are "delocalized" across the entire pi bond system and not "localized" in just one place.
To see how this would affect the acidity, compare the acidity of the -OH proton in ethanol to that of the -OH proton in acetic acid. In ethanol, the electrons on the oxygen have nowhere to go, and these electrons grip the proton attached to the oxygen pretty tightly as a result. On the other hand, in acetic acid the electrons on the -OH oxygen can flow towards the carbon to form a new double bond (sending the electrons that were present on the other double bond up to the other oxygen). This pulls the electrons away from the proton and makes it easier to remove. So acetic acid is more acidic than ethanol due to the resonance effe...sorry, mesomeric effect. M- pulls electrons away from the proton due to resonance and makes the proton easier to remove, while M+ pushes electrons onto the resonating system and makes the proton harder to remove.
A separate idea is to consider the effect of adding an electronegative atom to a single bond on the compound. This polar bond pulls on the electrons in a single (also called sigma) bond and can affect the acidity of the overall molecule in a similar way. For instance, compare the acidity of chloroacetic acid to regular acetic acid. Both compounds have an identical carboxyl group attached so that their acidity due to resonance would be expected to be the same based only on the mesomeric effect. However, the addition of a chlorine atom (I-) to the carbon next door adds sort of a long distance "pull" of electrons towards itself which amounts to a slight additional pull on the electrons in the resonanting caboxyl group away from the proton. This makes it even easier yet to remove the proton in chloroacetic acid and it is more acidic as a result. I- pulls electrons away from the molecule which ultimately corresponds to a further pull of the electrons away from the proton, increasing the acidity. I+ pushes electrons onto the molecule which ultimately pushes electrons towards the proton and makes it harder to remove.