Any function with a finite number of inputs can be specified by explicitly listing outputs corresponding to those inputs. This is what a truth table does. The notation $\Sigma(0,4,5,7)$ specifies a function that equals 1 for the following values of inputs: $a=0,b=0,c=0$; $a=1,b=0,c=0$; $a=1,b=0,c=1$; $a=1,b=1,c=1$. The values of inputs occur in rows 0, 4, 5 and 7 of the truth table if rows are counted from 0. Note also that the values of inputs: 000, 100, 101 and 111, are 0, 4, 5 and 7 in binary. For other inputs x(a,b,c) = 0 by definition.
Describing how to use Karnaugh maps is beyond the scope of this post. You can read about it, for example, in Wikipedia. There is also an online Karnaugh map generator here.