Space requirements for solving True Quantified Boolean Formulas problem
The main difference between time and space is that you can reuse space that is not needed anymore.

If you return from the recursive call that evaluates $A$, you can reuse the space used by this computation to compute the value of $B$.  Because of this you only need to store one branch of the decision tree of the original QBF formula in memory, which gives you a linear space requirement.

This fact is explained quite well in standard introductory text books on Complexity Theory.  You can have a look into Sipser's Introduction to the Theory of Computation, for example. On page 332 (3rd edition) he explains the space requirement for SAT, of which QBF is a generalization.