How to connect 1.5mm SMT alternating pins to 0.1" pin header?
That's just a custom pad spacing that the designer came up with to connect the two boards together. The staggered 0.75mm pitch connections may not match anything you can buy. The pads are going to be fragile, since they're just held on by adhesive (no through holes to act as rivets).  

That leaves you with two options as I see it- make up a transition board using one of the cheap online PCB sources .. or use thinner wires (maybe AWG 30 wire-wrap wire) that won't rip the pads off and put some strain relief on them, perhaps going to a chunk of perf board before heading off to your external circuitry.  

If you do the latter, you can stick down a chunk of perf board with the header on it first, then run (not too taut) AWG 30 wires to the pads (or where the pads go if you have ripped the pad off). It should be do-able in maybe twenty minutes to half an hour- there are fewer than 30 pins, it looks like. 

If you do the former, you can just match the pin layout with a board just like the one with the micro on it, and wire the pads to a 0.1" header. It will look really nice, but you'll have to wait a while for the board to arrive in the mail. 

P.S. The ribbons look a little messy, but you did well to get those boards out without destroying everything.