I don't understand your confusion.
The unit will output a value in the 176 area when stationary and this will increase or decrease if turning in the relevant axis - you have to turn it at 500 deg/s to see full-scale output - that's a very fast turn, so you aren't likely to see a big response to turning it by hand, but that's quite OK, its not a show-stopper like you seem to think.
To measure angle you have to integrate the output, like I said - have a look at the plentiful discussions of this here and elsewhere. To detect a 90 degree turn you'll then have to look at this integrated output over time and see if its varied enough in the timescale of interest.
Or you can look directly at the rate of turn to decide if there is current turning motion - this won't be a big signal with this gyro, a few counts.
Part of the issue is that you are looking at a 1.35V signal with an ADC set to 5V full range. You can get some more precision by using analogReference(EXTERNAL) and feeding 3V3 to AREF pin - then the ADC will have a 3.3V range.
I seem to recall that that breakout has differently-scaled outputs from standard, was it x4.5 or something? Those might help give stronger signal at slower rates-of-turn.