Fourier Transform and the Delta Function
First you need to understand that the crucial property of the delta function is that it picks a single value of a function when it gets integrated

$$
\int_{-\infty}^\infty f(x)\delta(x-a)\;dx = f(a)
$$

Using this property to calculate the inverse fourier transform of 
\$
\pi \left[\delta(\omega+\omega_0) +  \delta(\omega-\omega_0)\right]
\$
you get
$$
\frac 1{2\pi} \int_{-\infty}^\infty  \pi\left[\delta(\omega-\omega_0) +  \delta(\omega+\omega_0)\right] e^{j\omega t}d\omega
= \frac 12 e^{j\omega_0 t} + \frac 12 e^{-j\omega_0 t} = \cos \omega_0 t
$$
So in order to get a certain amplitude you have to multiply the delta function by some factor (weight), otherwise you get an amplitude of 1.

Since the amplitude of delta function is infinity by definition, the height is often used to indicate the weight.