How was is that the southern states that seceded from the union during the American Civil War were able to mobilize a force able to take on a professional army so quickly?
In all of the states, North and South, mobilization was a matter left to the states. Both Lincoln and Davis asked the states for volunteer regiments and the states used their longstanding militia laws to raise and equip state regiments which were then mustered into federal service.
The Union Army in 1861 was no more a professional army than was the Confederate Army.
In the first chapter of William Schouler's History of Massachusetts in the Civil War Schouler, who was the Adjutant General of the Commonwealth in 1861, described how Massachusetts mobilized. The basic process was the same in all the states. In fact, it remained the same until the Federal Militia Act of 1903.
https://archive.org/stream/ahistorymassach01schogoog#page/n17/mode/1up
C.P. Headley covers the same ground in Part II of his Massachusetts in the Rebellion.
https://archive.org/stream/massachusettsinr00head#page/n20/mode/1up