Simple explanation as to why certain computable functions cannot be represented by a typed term?
Given that you do not want to learn precise concepts, here is an intuitive explanation. In the discussion below "function" always refers to a function mapping natural numbers to natural numbers (possibly undefined at some arguments).

Any programming language which has


computable syntax and rules of evaluation, and
implements every total computable function


necessarily implements some partial functions.

To see this, suppose it were the case that every definable function in this language were total. Because the language has computable syntax, we can enumerate all definitions of functions (just enumerate all strings and filter out those that cause syntactic errors). Because the rules of evaluation are computable, the second assumption allows us to conclude that in our language we can define the total function eval(n,m) which evaluates the n-th definable function on m (essentially this is an mini-interpreter written in the language itself). But then the function

λ k . (1 + eval(k,k))


is a total definable function which is different from every total definable function, a contradiction.

The simply typed $\lambda$-calculus satisfies the first condition and it defines only total functions. Therefore it does not satisfy the second condition.

As far as "standing on your head" is concerned, for a strongly-normalizing $\lambda$-calculus it is fairly easy to provide a total function which is not definable in the calculus, namely the normalization procedure itself. It is not very important how fancy your strongly normalizing calculus is, it could be the polymorphic $\lambda$-calculus, or Martin-Lof type theory, or the Calculus of constructions. (Exercise: if you could implement the normalization procedure, you could implement eval above.)