How does the Early effect limit the voltage gain?
Voltage gain for a common emitter amplifier (for instance) is determined in part by the collector resistor. Put simply, the higher the value of collector resistor the more output voltage signal for a given input signal.

For an ideal BJT with no early effect, once the BJT is operating in its active region, the slope of \$I_C\$ versus \$V_{CE}\$ is zero: -



Image from here.

Now flip this around and ask what the collector output impedance of the ideal BJT is; it is infinite i.e. for any change in \$V_{CE}\$, the change in \$I_C\$ is zero. The non-ideal BJT situation looks more like this: -



Image from here. In other words the collector has a high (ish) output impedance but it's not infinite.

And, the early effect makes the slope more pronounced as the collector current rises: -



Image from here. So, early effect can be said to be a gain-limiting factor because this non-ideal resistance is in parallel with the real collector resistor.