The complex pair does not "cause" the oscillation.
They indicate there will be one.
Think of the complex plane this way.
A pole located along the x-axis is showing the speed of an exponential response to a disturbance input. The farther to the left, the larger the negative number, the faster the exponential decays. If you were in the right half of the plane, for positive values of x, the exponential would grow, not decay and you have instability. This is why the left hand plane is the stable region.
Now poles off the x-axis appear in pairs. If the pair is right on the y axis, that corresponds to a pure oscillation which never stops. No dissipation to drain the energy. It is a pure oscillator. The higher up the axis, the faster the oscillation.
If the poles move off the axis into the left half plane, say at (-1, 1) and (-1,-1), you now have a combination [by multiplication] of the oscillation and the exponential decay. It is a damped harmonic oscillator.
So the following points correspond to exponentially decaying oscillations which are ...
(-0.1, 0.1) and (-0.1,-0.1) very low freq, very slowly decaying
(-0.1, 1000) and (-0.1,-1000) very high freq, very slowly decaying
(-1000, 0.1) and (-1000,-0.1) very low freq, very fast decaying
(-1000, 1000) and (-1000,-1000) very high freq, very fast decaying
The X axis value indicates the exponential rate [negative is decay, positive is growth => instability]
The Y axis value indicates the oscillatory rate
Hope that helps