Concept of convolution
Convolution basically tells you how similar two signals are as one of them is shifted in the x-axis and reflected.  So consider taking your signal \$g\$ and shifting it by various amounts.  The convolution will have a peak when the two signals are mirror images across the y-axis.  In this case, they already are, so the peak convolution occurs with a shift of 0. 

The most important thing to understand is that the meaning of the x and y axes have changed from (time, value) to (shift, alignment).  

I've ignored the meaning of the y-axis magnitude.  When I've used the convolution before it's been with normalized signals so that the values ranged from 0 to 1, but that is certainly not always the case.