How does Schlage SecureKey Re-keying mechanism work

Making a wild assumption since we don't have enough information: I'm guessing that what you have is a grade-3 Schlage cylindrical privacy set of the sort that's available from every hardware store in the country. There are three pieces to this unit -- an outer knob with a hole in it, an inner knob with the thumbturn, and the latch mechanism.

The installation guides on the Schlage website are for one-side-keyed knobs, but the principle is the same -- in fact, the mechanism is close to being the same.

If you look at step 6 in that guide (sorry, I don't have that one image by itself), you will notice that on the inner knob with the turnpiece, they tell you to look into the knob and make sure the slot is horizontal. If you look closely, you'll also see that the flat piece which runs from knob to knob is horizontal.

That's what you got wrong. It is possible to assemble the lock with these pieces rotated 90 degrees -- but only if you rotate them in opposite directions, with the result that they're both at their limit and each keeps the other from operating. This also has the side effect of tying up the knobs in the locked position.

(This is a general hazard when assembling mechanisms that are only designed to operate through part of their rotation rather than being able to spin endlessly. The position of the parts in this cycle is known as their "timing". You got the timing wrong.)

Remove the knobs, and reinstall them with the flat bar horizontal and the thumbpiece turned to match it. That should cure the problem, unless you've managed to do something more weird than usual.

In your first picture the round disk looking objects on either side of the keyway are screw covers that are driven after the screws are tightened.

A strong sharp object, something like a chisel that you don't mind getting dull would do the trick. The pin that is in the screw head isn't very long so the cover will fall off readily after the cover starts lifting out. After the covers are removed, the screws are removed using a Phillips head screwdriver. The screws under the covers are #1 the screws in the latch are #2.