Representing $-2.5$ as a floating point number I am trying to understand the following:

For example the number $-2.5 = -1*1.25*2^1$ is strored as:
  $S = 1$, Exponent $= 1+127 = 128$, Mantissa = $0.25 $

This got me.  How do you connect all these numbers to yield $-2.5$ ?

If you are given a sign bit $S$, 127-offset exponent $e$, and mantissa $m$, you calculate the value as 
$$(-1)^S2^{e - 127}(1 + m)$$
So for your values $S=1$, $e=128$, and $m=0.25$,
$$(-1)^1 2^{128-127}(1 + 0.25) = -2.5$$
