In his article "'What' Hath Brisk Wrought? The Brisker Derekh Revisited," R. Moshe Lichtenstein argues that,
the basic change wrought by R. Hayyim was the refocusing of the
learning process upon the hard halakhic data that emerge from a sugya,
instead of upon the mechanisms by which they were derived. To state
the point in more technical terms, the Brisker approach shifted the
learner's interest from the talmudic discussion itself (shakla
ve-tarya) to the practical implications thereof (nafka minahs)...This
is the reason that Rambam occupies center stage in the Brisker orbit,
for it is he who distilled the talmudic conclusions into pure halakhic
form, systematically omitting any interpretation or mention of the
accompanying discussion, presenting us only with the halakhic hard
data without encumbering it with any explanations.
See there for more elaboration on this point.
Rabbi Menachem Genack makes a similar argument:
The Brisker derech involves conceptual categories. Halachic acts,
thoughts and objects are all conceptualized and placed in categories.
Once placed in the proper categories, the halachic and conceptual
characteristics of the phenomena become more clearly defined and
logically comprehensible. The Brisker emphasis on categorization
explains the well-known Brisker attachment to the Rambam. The Rambam
took the entire corpus of halachah and categorized all its elements in
a strictly rational, analytical order. This is the same conceptual
groundwork on which the Brisker derech is based.