Why randomness space must be significantly larger than the commitment space |R|>|C|? the picture is from https://youtu.be/IkNZWJFcfcU?t=236
For the hiding property we require that the commitment value provides no information about the message. In particular we hope that for any given message $m$ it is possible for $H(m,r)$ to take all possible values in $C$ (otherwise one could exclude some messages as corresponding to some commitments). If $H$ behaves like a random function, then it unlikely to have this surjective property unless $|R|>|C|\log|C|$.
More powerfully we probably want the values of $H(m,r)$ for any fixed $m$ to be distributed uniformly across the values of $C$.
Note that these are information theoretic "zero knowledge" requirements rather than complexity/computation bounded requirements.