A typical flyback design

From my understanding, if we go above the primary peak power, the
transformer will saturate. How this CCM mode overcome that?
It's better to talk about reaching a peak in primary current rather than a peak in primary power because it's an excess of current that creates core saturation.
So, if you define \$I_{MAX}\$ as the maximum allowable current to avoid excessive core saturation then, the maximum energy you can extract from that primary is when the secondary current has fallen from maximum to zero amps i.e. all the magnetic energy is removed from the core. Below are a couple of example waveforms of primary and secondary current: -

The lighter load (red and green solid lines) is dispensed less energy per switching cycle and naturally, there is a "hold" in the latter part of that switching cycle. The full load scenario is represented by the dotted lines i.e. the "hold" period is zero and the peak current rises to \$I_{MAX}\$.
So, you can’t rise higher than the saturation point (\$I_{MAX}\$) and, you can’t deliver maximum energy in a single switching cycle unless secondary current (and flux) falls to zero. This is the boundary between CCM and DCM.
Based on that, does this means that this peak power can be achieve
without changing the size of the transformer?
If the operating frequency and core are fixed then, this is the maximum power that can be achieved. However, if the switching frequency were (say) doubled, the flyback converter could run at higher power using CCM; the peak curent would still be \$I_{MAX}\$ and it would fall to \$I_{MAX}/2\$ before the next switching cycle began.
The energy dispensed each cycle would be only 75% of what it was when operating at the previous switching frequency but, the power dispensed would be 2 x 0.75 of what was previously shown. So, by doubling the switching frequency, the power throughput is 50% greater in CCM than in DCM at half the switching frequency.
But, if you can "run" at twice the switching frequency, then why not design the flyback transformer to run in DCM with lower primary inductance?
So, if you halved the primary turns you get: -
- Primary inductance falling by 4.
- Peak current can rise to \$2\cdot I_{MAX}\$ for the same core saturation levels.
The impact of these is that the energy stored in the first part of the switching cycle is now: -
$$W = \dfrac{1}{2}\dfrac{L}{4}\cdot [2\cdot I_{MAX}]^2 = \dfrac{1}{2} L\cdot I_{MAX}^2$$
And, this is exactly the same energy as was stored in the original DCM design when running at the original frequency but now, because the switching frequency has doubled, that energy is converted twice as often and, twice the power is dispensed into the secondary load.
If you refer back to what I wrote earlier for a CCM design operating at twice the switching frequency, the power improvement is only 1.5.