In my lecture notes we have the following:
Proposition: $$V(S)=V(\langle S \rangle )$$
Proof:
$$\langle S \rangle=\left \{\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in R=K[x_1, x_2, \dots , x_n]\right \}$$
It stands that $S \subseteq \langle S \rangle$
$\Rightarrow V(S) \supseteq V(\langle S \rangle )$
Let $\overline{a}=(a_1, a_2, \dots , a_n) \in V(S)$
$\Rightarrow [f(\overline{a})=0, \forall f \in S]$
Let $h \in \langle S \rangle$,
$h=\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in S$
So $h(\overline{a})=\sum_{i=1}^m g_i(\overline{a}) f_i(\overline{a})=0$
$\Rightarrow \overline{a} \in V(\langle S \rangle )$
That means that $V(S) \subseteq V(\langle S \rangle )$
Finally, $V(S)=V(\langle S \rangle )$.
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Why do we write $\langle S \rangle$ as $$\langle S \rangle=\left \{\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in R=K[x_1, x_2, \dots , x_n]\right \}$$ ???
Why is the upper limit of the sum $m$ and $n$ ???
Why does it stand that $S \subseteq \langle S \rangle$ ???
These are general facts about ideals.
The ideal generated by $S$ is either by definition or as one of the first consequences of the definition (depending on how things are set up) the set of all finite $R$-linear combinations $\sum_{i=1}^m g_i f_i$ with $g_i \in R$ and $f_i \in S$. This is the smallest ideal that contains $S$. An analogy would be the vector-subspace generated by a set.
The $m$ has no particular meaning. It is just some variable to write down the sum. It can also change from one sum to the other. You can replace it by whatever you like. The important thing is to consider all finite sums of elements in $S$. In case the set $S$ is finite it can be natural to choose $m$ the cardinality of $S$ and to index the elements of $S$. However this does not work in case of infinite $S$. An alternative is to write this as $\sum_{f \in S} g_s f$ with $g_s \in R$ all but finitely many $0$.
For $f \in S$ one has that $f= \sum_{i=1}^1 1 \cdot f \in \langle S \rangle $.