Please compare the following two definitions:
Definition 1
A representation of a topological group $G$ in a vector space ${\mathbb{V}}$ over complex numbers is a continuous map $$ A\, :\quad G\times {\mathbb{V}}\longrightarrow{\mathbb{V}}\;\,,\qquad (g\,,\;v)\longmapsto A(g)\,v\;\,,\;\;\;\;\; g\in G\,,\;\;v\in{\mathbb{V}}\;\; \label{1}\tag{1} $$ with the following properties:
(1) $\;\; A\,$ is a group action;
(2) $\;\; A\,$ is linear, i.e. $$ A(g)\,(\alpha\, v+\beta)=\alpha\, A(g)\, v+ A(g)\, w\quad \mbox{for}\quad\forall g\in G\,,\;\; v,\; w\in {\mathbb{V}}\,,\;\; \alpha\in{\mathbb{C}}\,\;. $$
Definition 2
A representation of a topological group $G$ in a vector space ${\mathbb{V}}$ is a homomorphism of $ G$ into the group of invertible linear transformations of ${\mathbb{V}}$: $$ A\,:\quad G\longrightarrow GL({\mathbb{V}})\;\,. \label{2}\tag{2} $$
$$ \, $$ Definition 2 $\;\;\Longrightarrow\;\;$ Definition 1,
Indeed, the $A$ from Definition 2 is linear and ensures $\,(g\,,\;v)\longmapsto A(g)\,v\,$.
Also, Definition 2 says that $A$ is a homomorphism $-$ which guarantees the continuity of $\,(g\,,\;v)\longmapsto A(g)\,v\;$ in Definition 1.
QED
To show that Definition 1 $\;\Longrightarrow\;$ Definition 2, we must demonstrate that Definition 1 ensures \eqref{2} being a homomorphism.
How to do this?
Actually, a group action by $G$ on a set $X$ is equivalent to a group homomorphism $\psi: G\to \operatorname{Aut}(X)$, where in this case automorphisms are bijections of the set with itself. Indeed, the group action can be thought of as $\alpha:G\times X\to X$ such that $\alpha(hg,x)=\alpha(h,\alpha(g,x))$ etc. This is equivalent to $\psi(hg)=\psi(h)\circ \psi(g)$ where $\psi(g)(x)=\alpha(g,x)$. $\alpha(e_G,x)=x$ for all $x\in X$ is equivalent to $\psi(e_G)=\operatorname{Id}_X$. These are good facts to check carefully yourself as an exercise in the definitions (also usually in any algebra text).
So, definition (1) already gives us a group homomorphism $G\to \operatorname{Aut}^{\operatorname{Sets}}(V)$. A priori, the codomain on the right is self-bijections. However, the requirement that $G$ act linearly means that $G$ maps into $\operatorname{GL}(V)\subsetneq \operatorname{Aut}^{\operatorname{Sets}}(V)$. The continuity of the action stipulated in definition (1) then implies that as a map $G\to \operatorname{GL}(V)$ it is continuous, where $\operatorname{GL}(V)$ is viewed as a topological group.