Syntax of a Pushdown Automata transition function
The rule, in English, can be rendered roughly as follows:


  If the machine is in state $q$, and $S$ is the topmost stack symbol, the machine may do either of the following things without consuming any input: it may remain in state $q$ and replace $S$ with $aaB$; or it may remain in $q$ and replace $S$ with $bbA$.


A good way to think about PDAs and transitions is to reason about configurations. The configuration of a PDA consists of the following information: the state, the unread input, and the contents of the stack. Indeed, this makes the transition function (almost) a (partial) function from configurations to (sets of) configurations.


$q$ is the state;
$\lambda$ is used to indicate that the current symbol may be input symbol, and that the input should not be reduced as it normally would after a transition;
$S$ gives the topmost stack symbol, which is the only part of the stack contents the PDA can see.