How does "δ:Q×Σ→Q" read in the definition of a DFA (deterministic finite acceptor)?
$\delta$ is a function which takes a current state and a letter from the alphabet as arguments and produces next state.

It might be easier to understand when you realize that it is just a function that takes a pair, for instance $(q_0, a)$ where $q_0 \in Q$ is a state and $a \in \Sigma$ is a letter in the alphabet, and $\delta(q_0, a) = q_1$ means that if in state $q_0$ you read an 'a', proceed to state $q_1$.

It is the definition of a transition function which is used for example in Turing machines.

$\times$ means Cartesian product and $\to$ means "produces" or "maps to".