How to introduce type theory to newcomer?
Some remarks to be pointed out at the beginning:
Type theory should be thought of as a general mathematical theory of constructions: how mathematical objects are constructed, what are different kinds of constructions (functions, pairs, inductive constructions, co-inductive constructions, etc.), what equational laws they satisfy, and how they can be generally handeled. Thus we should expect a systematic way of introducing new kinds of constructions. Indeed, as we shall see every new way of constructing types follows the same path: formation rules, introduction rules, elimination rules, congruence rules, and equations which explain how the introduction and eliminations fit together.
Classical mathematics rests on two pillars: first-order logic and set theory. Set theory is formulated using first-order logic. In contrast, type theory stands on its own. There is no logic beneath type theory. In fact, we can see type theory as a common unification of logic and set theory: certain types are like logical propositions, some types are like sets, and there are also types which are like nothing they have seen before.
Because there is no logic, only a formalism for constructing stuff, one has to get used to a new way of speaking and thinking. In type theory it is impossible to say or think "$P$ is true" because the only mechanism available is construction of an element of a type. Since $P$ is a (special kind of) type, we should instead say "we construct an element $e$ of type $P$". (In first-order logic the element $e$ would correspond to a proof of $P$.)
Therefore, since in type theory it is impossible to state that a proposition $P$ holds without actually giving a proof $e$ of it, proofs are unavoidable. Or to put it in another way, proofs are "first-class citizen", as they are just elements of certain types. The situation in ordinary math is quite different: even though proofs are central to mathematics, nobody ever considered the set of all proofs of a propostion $P$, or anything like that.