In the following question and answer I kind of get it how it is calculated at surface but I don't get it fully. Can someone please explain the answer in depth?
The answer to your question is essentially contained in Dilip's comment, but I guess as a beginner you need some more help. First, it's important to realize that the response of an LTI system to a scaled and shifted impulse $$x[n]=a\delta(n-k)$$ is simply a scaled and shifted version of its impulse response:
Furthermore, the response to a sum of signals equals the sum of the responses to the individual components of the input signal. Consequently, the response to the given input signal
Now you just have to express the impulse response $h[n]$ in terms of $\delta$-impulses. By noting that $$u[n]-u[n-3]=\delta[n]+\delta[n-1]+\delta[n-2]$$ you get
Plugging (2) into (1) gives the desired result.