Why does the temperature rise?

The formation of covalent bonds between H⁺(aq) and OH^–(aq) is more exothermic than the breaking of the forces between the ions and the water molecules is endothermic, with the net result that the reaction gives out heat, causing the temperature to rise.

What is the significance of the maximum temperature?

The maximum temperature occurs at the point of neutralisation. The alkali has been used up and water has been formed in an exothermic reaction. After this point, more acid is added but there is no more alkali, hence no further reaction occurs and no more heat is given out. The addition of further acid (the acid is at room temperature) after neutralisation only cools the solution. The temperature decreases.

Why would this method be inaccurate for a reaction between an acid and a solid base?

The reaction between an acid and an alkali occurs as soon as the solutions are mixed. It is, in effect, instantaneous. A slower reaction would not have released all its heat as quickly, so heat loss would be much greater because you would need to wait for the reaction to finish each time.