The original post has a misunderstanding that is quite common. It says
But the range does not start at 192.168.1.1. The range is actually 192.168.0.1 through 192.168.1.254
If we want to get into the details of it then think of the third octet in binary. The mask of 254 is 11111110 which indicates that the first 7 bits of addresses must match and the last bit is variable. Then think of the address in binary which is 00000001. Change the last bit and the other available address is 00000000. So the range begins at 192.168.0 and goes through 192.168.1.