Unit Impulse funciton FT
If follows directly from the definition of the Dirac delta distribution. It is defined so that

$$\int_\mathbb{R} \delta(x) f(x) dx := f(0)$$

for any test function $f(x)$. In other words, the Dirac distribution is the generator of the linear functional that extracts a single function value.

With this definition the Fourier transform of the Dirac distribution is simply:
$$\int_\mathbb{R} \delta(t) \exp(-2\pi i \omega t) dt=\exp(-2\pi i \omega\cdot 0)=\exp(0)=1$$

It doesn't make much sense to say the Dirac distribution is infinitely high or infinitely narrow. Just use the definition given above and apply it. If you really need to, you can understand the distribution as the limit of a sequence of certain functions, but it's not a function itself. And specifically it doesn't have a graph.