Let us be given a linear algebraic group $G$ over a field $K$ of characterstic zero. This group $G$ is defined as the common zeroes of a finite set of polynomials $\{f_1, \ldots ,f_r\}$ $\in K [T_1,\ldots,T_n]$with some additional properties like having the structure of a group along with 2 morphisms of varieties.
Now, the tangent space of $G$ at a point $x$ can be thought of as common zeroes of $\{d_xf_1,\ldots,d_xf_r\}$ where $d_xf= \sum_{i=1}^{n} \frac{\partial f}{\partial T_i}(x) (T_i - x_i)$. Let us think of the $Lie(G)$ as the tangent space at $e$ defined this way.
Now, suppose we are given a group action of $G$ on $G$, say $$\phi : G \times G \to G$$ sending $$(g,x) \mapsto gxg^{-1}$$
How does this map induce a represention :
$$\psi : G \to GL_{Lie(G)}$$
Please help me understand in terms of the common zeroes as how the common zeroes are mapped by an element of $G$. I have just started to read algebraic geometry. So, an intutive explanation will be really helpful.