How many bits needed to describe completely random function?
The word random is misplaced here, the question is how many bits are required to describe a function from $U$ to $\{1,...,n\}$. The connection to randomness here is simply the fact that you need to deal with an arbitrary function (since you pick one uniformly at random). Thus, you cannot rely on any structure of the function, which could allow a more compact description (e.g. a constant function would require only $\log n$ bits to describe).

One way to look at it, is saying that you need to specify the output for every possible input. Since there are $|U|$ possible inputs, and specifying an element in the range requires $\log n$ bits, this results in a $|U|\log n$ length description. You can also use the fact that there are $n^{|U|}$ different functions from $U$ to $[n]$, so you need $\log n^{|U|}=|U|\log n$ bits to index an element in this set.