How does dsDNA become ssDNA when binding to a nitrocellulose membrane?
The DNA is denatured in-gel prior to transfer using alkali conditions, most commonly a solution containing 0.5 M NaOH (buffered with 1.5 M NaCl). Following that the pH of the gel is neutralised in 0.5 M Tris-HCl, pH 7.5; 1.5 M NaCl. 

I have found the Roche DIG application manual to be a well-written and reliable resource for performing Southern and Northern blots. The section on Southern blots starts at page 94 and you can modify the protocol to suit other detection systems.