The contactor isn't working because the coil (which pulls in the little button) isn't connected to anything as far as I can see in your picture.
See the lable on the right side of the box? (The one that says read before wiring LOL), under it it says 'coil', that's to tell you what screws yto use for the coil)
There are a pair of terminals on it somewhere , you are supposed to supply power to the coil through a pressure switch (and on some compressors through a low oil p cutout) to activate the coil which in turn makes the motor run.
If you have the pressure switch wired in somewhere else (sounds like it) you can just jump the coil to the power to make it always on, tho it's kinda a weird thing to do...(from the web site, furnas/siemens contactor 42bf- is 220v , but I can't read the number in your photo all that well)
>I dont see where there is a fusible link
Those 3 tin colored vertical strips of metal with a screw at each end are thermal motor fuses ...
Oh, and the little red rectangle button is the cover safety switch, because you are pushing the contactor in by hand, it's not going to do anything, but when you have it wired right, it prevents the contactor from operating with the door open.
The ground should NEVER pass through a contactor, and it shouldn't be on a lug with an insulated wire...
Is the right most wire a ground or a neutral?
If it is a ground, it should pass through the box, and be attached to the box, not to anything on the contactor assembly.
If it is a neutral, it shouldn't be attached to the box, and you need *another* wire for a neutral, but, I don't think you need a neutral for this thing, it's 220, with nothing else that needs 110 , so you don't need a neutral to get 110, but you *do* need a ground somewhere....
(Confused yet? Sorry, I don't know how ot make this simple....:(