After more than 2 years in lurk mode, just had to chime in and mention that flat-earther/geocentrist bullshit is a new low point for NBR.
Having said that, if you would like an actual, ridiculously over simplified answer for your question: Yes, The atmosphere 'spins' with the planet, it's a fluid and "An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. " - Newton's First Law. We can measure the rotational speed of the earth using a Foucault pendulum or a ring laser gyroscope, it's accurate and repeatable. So the very fact we don't experience the violent '1000 mph winds' (at the equator) you're talking about IS confirmation of a 'spinning atmosphere'.
A more lengthy explanation involves inertial frames, fictitious forces, fluid dynamics (because the atmosphere is a fluid) and about a dozen other concepts which require a great deal more physics knowledge than I possess. For simple physics, we can treat the earth and it's surrounding atmosphere as an inertial frame, so the only thing we really need to account for in Baumgartner's jump is local wind conditions around his capsule as he ascended. So, he landed 38 miles away because as he ascended he experience, on average, a 15 mph wind.
Expressing the rotation of the Earth as speed can also be problematic, because how far north or south of the equator you are, and your elevation greatly effects that rate. (i.e Someone standing at the north or south pole would be travelling zero mph, while someone at the equator would be travelling 1000ish mph). Elevation is also a factor. So, expressing it as angular velocity or revolutions per minute makes more sense, the earth rotates at .000694 RPM, not 1000mph.
It should also be said that just because something can't be explained to a 'simpleton', doesn't mean that it's not true, it just means that the simpleton doesn't have the requisite knowledge to understand the explanation. Complex systems generally require complex explanations.
Any-who, back to lurk mode.