Why is the number of ways of choosing $0$ items from $n$ items $1$? It seems easy to grasp that number of ways of choosing $n$ items from $n$ items is 1. But I am unable to understand why is it 1 for choosing 0 items.

When you phrase it in plain English, the answer isn't necessarily totally clear. It might seem reasonable to argue that there are $0$ ways of choosing $0$ things from $n$, since choosing $0$ things isn't really a choice. However, when mathematicians talk about "the number of ways to choose $0$ from $n$," we mean something a bit more specific. 
A few ways of describing what we mean:


*

*The number of subsets of an $n$-element set with $0$ elements (and we always assume the empty set counts)

*The number of functions from a $0$-element set to an $n$-element set, up to permutations of the domain (there's exactly one function from the empty set to any set).

*The coefficient of $x^0y^n$ in the expansion of the binomial $(x+y)^n$. 


All of these agree that there is one way to choose $0$ out of $n$ things, and all of these perspectives are mathematically very useful. Thus the perspective that there is a way to choose $0$ out of $n$ things is universal in mathematics. 
This wasn't always so obvious to everyone: for instance, I have heard of an early 20th century mathematician who insisted in his books that all intersections be nonempty to be defined. It seems to me that this would be consistent with refusing to accept the empty set as a subset, and with saying there are $0$ ways to choose $0$ things from $n$. But this would make for lots of inconvenient circumlocutions, so we've settled pretty firmly on the other solution.
