How do you say $\delta\colon Q \times \Sigma \to Q$ in English? Describing what $\times$" and $\to$ mean would also help.
$\delta$ 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".
The statement $\delta\colon Q \times \Sigma \to Q$ translates to: The function $\delta$ assigns every pair formed by an element of $Q$ and an element of $\Sigma$, an element of $Q$. Instead of "assigns" you could also say "maps to". I assume that $\delta$ is the transition function for a finite automaton. Then you can say, $\delta$ assigns every pair, formed by a state and character a new state.
Here is a short explanation how to interpret the symbols you are asking for:
The statement $\delta: Q\times\Sigma\mapsto Q$ can be read as follows: given a state and a symbol the automaton moves to another state (possibly the state it was in). The $\times$ is like "and" in English meaning that you can pick any element from the first set $Q$ and any element from the second set $\Sigma$. More precisely, it's called the Cartesian product of the sets $Q$ and $\Sigma$. You can think of it as an operator that takes sets and outputs a new set that is the combinations of elements from $Q$ and $\Sigma$. Here is a simple example:
The symbol $\mapsto$ assigns elements taken from the right set to elements taken from the left set. You can think of it as a 2-column table, here is an example:
$(q_0,a)$ maps to $q_1$ and so on.