Overall Reaction
This reaction is identical with the acetoacetic ester synthesis, except that we bein wth diethy malonate and end with a carboxylic acid. Otherwise the chemistry is identical.
Enolate formation and Alkylation - malonic ester synthesis
In the first reaction, the diethyl malonate is deprotonated to form an enolate. The enolate is then treated with a primary alkyl halide undergoing an SN2 reaction with the enolate. In these steps we made a substituted malonic ester. Hence this is referred to as the malonic ester synthesis.
Decarboxylation
If desired the substituted malonic ester can be easily decarboxylated to form a ketone. Simply heating the ester in acidic water will decarboxylate the malonic esters.
Under these conditions, the ester is first hydrolyzed (acyl substitution) and then decarboxylates. The decarboxylation is a pericyclic reaction (e.g. Diels Alder and Claisen Rearrangement were the other pericyclic reactions you may have seen in Ochem). In the end, a substituted carboxylic acid is formed.