  The discovery of Fractional Chern Insulators (FCIs) in twisted bilayer
MoTe$_2$ has sparked significant interest in fractional topological matter
without external magnetic fields. Unlike the flat dispersion of Landau levels,
moir\'e electronic states are influenced by lattice effects within a
nanometer-scale superlattice. This study examines the impact of these lattice
effects on the topological phases in twisted bilayer MoTe$_2$, uncovering a
family of FCIs with Abelian anyonic quasiparticles. Using a composite fermion
approach, we identify a sequence of FCIs with fractional Hall conductivities
$\sigma_{xy} = \frac{C}{2C + 1} \frac{e^2}{h}$ linked to partial filling
$\nu_{\,\text{h}}$ of holes of the topmost moir\'e valence band. These states
emerge from incompressible composite fermion bands of Chern number $C$ within a
complex Hofstadter spectrum. This approach explains FCIs with Hall
conductivities $\sigma_{xy} = (2/3) e^2/h$ and $\sigma_{xy} = (3/5) e^2/h$ at
fractional fillings $\nu_{\,\text{h}} = 2/3$ and $\nu_{\,\text{h}} = 3/5$
observed in experiments, and uncovers other fractal FCI states. The Hofstadter
spectrum reveals new phenomena, distinct from Landau levels, including a
higher-order Van Hove singularity (HOVHS) at half-filling, leading to novel
quantum phase transitions. This work offers a comprehensive framework for
understanding FCIs in transition metal dichalcogenide moir\'e systems and
highlights mechanisms for topological quantum criticality.
