What does the leading turnstile operator mean?
On the left of the turnstile, you can find the local context, a finite list of assumptions on the types of the variables at hand.

$$
x_1:T_1, \ldots, x_n:T_n \vdash e:T
$$

Above, $n$ can be zero, resulting in $\vdash e:T$. This means that no assumptions on variables are made. Usually, this means that $e$ is a closed term (without any free variables) having type $T$.

Often, the rule you mention is written in a more general form, where there can be more hypotheses than the one mentioned in the question.

$$
\dfrac{
  \Gamma, x:T_1 \vdash t : T_2
}{
  \Gamma\vdash (\lambda x:T_1. t) : T_1\to T_2
}
$$

Here, $\Gamma$ represents any context, and $\Gamma, x:T_1$ represents its extension obtained by appending the additional hypothesis $x:T_1$ to the list $\Gamma$. It is common to require that $x$ did not appear in $\Gamma$, so that the extension does not "conflict" with a previous assumption.