The term 'functional' in this context implies a type of analysis that is able to provide a mechanistic understanding.
The authors first describe their exploratory approach, by which they identified some genes and their mutations that are associated with patient relapse in acute lymphoblastic leukemia. Then they proceed to intervene in the system, by expressing mutated KRAS in mice. When they did that, they observed several hallmarks of relapse in those mice. Hence they can conclude that a mutation of this gene is involved in the mechanism that underlies relapse.
Thus the way these authors use the term 'functional analysis'/'functional dissection' is in contrast to a correlational approach.
The underlying assumption is that by isolating parts of the system and studying their properties and their effects we will be able to understand the physical world. This is a philosophical (metaphysical) thesis called reductionism, which is arguably the prevailing view in the natural sciences.
The first use of the term "functional dissection" I could find is in a 1971 paper on the immunological response of small molecules. Their goal in that paper was to dissociate (or dissect) the effects of small molecules, such as a bovine glucagon, on antibody specificity versus antibody production. They have found that different parts of this molecule were responsible for antigen specificity (all antibodies recognised the N-terminal) and cellular immunity (mostly the C-terminal of the molecule was associated with DNA synthesis).