When a multimeter is measuring the value of the device under test (DUT), it is doing this many times a second. Better meters make more measurements per second than lower quality meters. The problem with these individual measurements is that each of them can be slightly different from each because of ambient electrical nose, unsteady hands on the probes making intermittent contact, and the inherent characteristics of the device which also might be a bit noisy.
To help give a more stable and easier to read value on the display, the meter will average many of the measurements before displaying the value. In the case of the ohms function, when the probes are not contacting anything the measurement is infinity. The meter will accumulate a number of these infinity measurements and then display them. When you touch the probes to the DUT it see first one measurement of the DUT plus the other averages of infinity. So the first displayed value will be just under the maximum range of the meter, then with each successive measured value being added to the average the displayed value drops to the real value. When all the averages are actually from the DUT, that is when you get the stable reading you expect.