This answer answers the title "Why does spinning help stabilisation" rather than specifically for launches.
Before jumping into the answer it is worth looking at the background. With rigid bodies then adding a spin will help average out asymmetries. However neither rockets or satellites are rigid, they contain liquids and flexible appendages and these are both means of energy damping, and means by which energy can transfer from one axis of rotation to another.
The basic ideas of stability in this context (stability is a big subject, even in maths before one tries to apply it to any real object) are that objects that spin can be stable if they spin around their axes of maximum or minimum inertia. Good examples would be the central axis of a coin (like a windmill) and pencil (like a barbeque) respectively.
This doesn't hold for bodies with energy transfer modes from propellant sloshing and appendages. A pencil like rocket spinning longitudinally can, through flexible modes, transfer all of its rotational energy into a rotation about its axis of maximum inertia. This means it will adopt an "end over end" motion. Please note I'm not claiming that its angular momentum has changed here. There is a case of this really happening with an early small launch vehicle (late 50's I think).
If there are flexible modes then the only rotation that is stable will be about the axis of maximum inertial, i.e. a windmill rotation rather than barbeque mode.
Spin stabilisation is used on many satellite designs though in every case the designers have to pay close attention to the shape of the satellite so that it is behaving firmly in windmill mode, i.e. spinning around its axis of maximum inertia.
Extra detail (EDIT)
The early small launch vehicle I mentioned was the Explorer 1 / Juno 1 mission the first US satellite. It was intended to spin at 750 rpm along the long axis but was found to be spinning at 7.5 - 8 rpm which corresponded to the energy having transferred into one of the transverse axes. Its well known enough to be a class room dynamics example. This image below shows the whip antennas which are, I believe, the physical mechanism for the energy transfer between axes.