From Tao's Analysis I:
I'm having trouble parsing this {...} expression. I'm not sure about the operator/quantifier precedence, and I'm especially unclear on how to read, and understand, "$(x_\alpha)_{\alpha\in I}$". How do you write this whole expression in prose? How do I make sense of $(x_\alpha)_{\alpha\in I}$?
In the case of an infinite cartesian product $\prod_{\alpha \in I} X_\alpha$, the elements of the set are functions $$f:I \to \bigcup_{\alpha \in I} X_\alpha$$ which satisfy $f(\alpha) \in X_\alpha$ for every $\alpha \in I$, notice that this is exaclty the definition given by Tao, since $$ \left(\bigcup_{\beta \in I}X_\beta\right)^I $$ is just the set of functions from $I$ to $\bigcup_{\beta \in I}X_\beta$. It is common to denote one such function $f$ by $(x_\alpha)_{\alpha \in I}$ in analogy with the tuple notation for finite products.
A good way to make sense of this definition is to think of an element of the cartesian product as a way to choose one element from each set $X_\alpha$, where the index $\alpha$ runs over $I$. Then the function $(x_\alpha)_{\alpha \in I}$ is like the rule telling you how to pick the elements.
This might seem strange at first, but it relates in a nice way with the case of finite products. For example, in this definition, the element $(7,\pi) \in \mathbb{N} \times \mathbb{R}$ is actually thought of as the function $f:\{1,2\} \to \mathbb{N} \cup \mathbb{R}$ defined by $f(1)=7$ and $f(2)=\pi$. Following the notation of the definition, we have $I=\{1,2\}$, $X_1=\mathbb{N}$ and $X_2=\mathbb{R}$.
If you are familiar with the Axiom of Choice, another possible formulation of it is by saying that an infinite cartesian product is non-empty, or following the intuition above, given an infinite family of sets, it's possible to define a function which chooses one element from each set of the family.