In relativity we measure distance using a reference frame, and generally this is our rest frame i.e. the frame in which we are at rest at the origin. So in my rest frame here on Earth I measure the distance to you in your rocket as $\ell_0$.
Since you and the rocket start off at rest relative to me we share the same reference frame, except that we may disagree where the origin is, so you also measure the distance between us to be $\ell_0$.
Lorentz contraction happens because the reference frame used by a object moving relative to me doesn't line up with my reference frame. Specifically we disagree about where the time and distance axes are so what looks to me like a distance will look to you like a combination of distance and a time, and vice versa.
Now back to your question about the rocket. When you and the rocket start accelerating my reference frame doesn't change, and none of the distances measured by me in that frame change. So the distance from me to you is still $\ell_0$, apart of course from the fact it is now decreasing with time because you're moving towards me.
But your reference frame has changed. You started in a frame at rest with respect to me, and now you're in a frame moving at speed $v$ with respect to me. So the distances you measure will have changed, and your measurement of the distance between us will have decreased to $\ell_0/\gamma$.
But how do we know that you are the one who changed frames and not me? Well that's because you are the one who accelerated i.e. you felt the g-forces as your rocket motor cut in. In special relativity it is unambiguous who accelerates and who doesn't, because all you have to do is measure the g-forces you feel. Sitting here on Earth I felt no g-forces so I know that I haven't accelerated and my reference frame hasn't changed.