  Charge order in kagome metals is of extensive current interest. Recently, a
charge density wave was discovered in the magnetic binary kagome metal FeGe. In
analogy to its predecessor, the non-magnetic $A$V$_3$Sb$_5$ ($A$=K, Cs, Rb),
the in-plane ordering occurs at the $M$ point. In contrast, however, the system
manifestly shows effects of substantial correlations. Here we identify the
topological bands crossing the Fermi energy (E$_F$) in FeGe and characterize
the correlation-induced renormalization of these bands. We then derive a charge
order from an effective model comprising topological kagome `flat' bands in the
presence of a magnetic order. We demonstrate edge states as well as excess
out-of-plane magnetic moment associated with the charge order; both are
fingerprints of non-trivial band topology and consistent with recent
experimental observations. Our results point to FeGe as an ideal platform to
realize and elucidate correlated topological physics.
