Why shaking the electron faster require more work ?

Because the electron has to push against the field that it just created. If you shake it slowly, there is time for that field to propagate away.

@An_Elephant Shaking your fist does too. Remember power is per time. Faster shaking = more changes in direction (acceleration) per time.

@user1402154 So the field created by electron push the electron itself? Means electron is exerting force on itself to oppose my shaking?

@DKNguyen Faster shaking = more power (perhaps) . Faster shaking = more work? How?

@An_Elephant More power = more work/energy over the same time interval. If you don't understand that you need to go back and study some basics.

Shaking the electron faster would require more work, but this does not mean each photon has to carry more energy. It's possible to postulate a model where such electron would generate a greater number of less-energetic (and yet high-frequency) photons. In this model "high frequency has high energy" does not stand, but your answer ("shaking the electron faster (at a higher frequency) would require more work") is equally "good". The problem with this model is it does not match experiments. Since your answer can "explain" the right model and the wrong one, I say: non sequitur.

@An_Elephant yes, en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force

each individual shake period is the same work, I think? So, more shakes per second, more work per period of time

@user1402154 Wow, I never knew about such.

@DKNguyen No, I know this.

@An_Elephant Very good.

@DKNguyen I now understand a little that how more work is done and higher energy wave is produced by more shaking. But why more shaking or say higher frequency electron directly implies higher frequency EM wave?

@An_Elephant That's best answered by a physicist which I am not.

This is dead wrong. The reason high-frequency photons carry more energy is exclusively quantum mechanical, and no amount of classical intuition is going to explain it. To be clear, within classical mechanics, higher-frequency shaking does not imply more work, since you're not specifying the amplitude of the shaking.

@Paul Given the position of this answer (high score on a "Hot Network Question", which gets advertised throughout stack exchange), it is now effectively acting as misinformation. I appreciate that this is unintentional, and the runaway effect of an answer which is intuitively appealing is hard to control. But this answer is doing more harm than good right now. Please consider a substantial edit, or other such measures.

I agree with @EmilioPisanty. This answer confirms the classic intuition, which is usually wrong when dealing with QM. Keeping this answer at the top will spread misinformation on top of confirmation bias.

The answer correctly says that shaking the electron faster requires more power, but this has nothing to do with the quantisation of that power into photons per second. The equation for the energy of a photon E = hf has nothing to do with the total power delivered.

Therefore does the electron movement have a large bearing on the issue of energy?

rephrased: doesn't the E=hf hold even for a photon in a vacuum?