By definition, a direct limit of a system $\{B_i\}_{i \in I}$, if it exists, is an object $\varinjlim B_i \in \mathsf{C}$ together with morphisms $f_i : B_i \to \mathcal{B}$, such that for all objects $A \in \mathsf{C}$,
$$\hom_{\mathsf{C}}(\mathcal{B}, A) \cong \hom_{\mathsf{C}^I}(\{B_i\}, |A|) \tag{*}$$
is a bijection, the map being given by
$$(g : \mathcal{B} \to A) \mapsto \{ g_i = g \circ f_i : B_i \to |A|_i = A \}_{i \in I}.$$
You can easily see that this is equivalent to the universal property of the direct limit: it exactly means that for all system of morphisms $g : B_i \to A$ (compatible with the system $\{B_i\}$) there exists a unique morphism $\mathcal{B} \to A$ compatible with the $g_i$. (It's one of these things you need to work out yourself to really understand, so I encourage you to write down the details.)
It can be shown that if for all systems $\{B_i\} \in \mathsf{C}^I$, the direct limit exists, then the constant direct system functor $|-| : \mathsf{C} \to \mathsf{C}^I$ has a left adjoint. It can be denoted $\varinjlim : \mathsf{C}^I \to \mathsf{C}$ and is called the direct limit (or colimit) functor. Since this is a left adjoint, equation $(\text{*})$ holds (with $\mathcal{B} = \varinjlim B_i$), and you see that therefore $\varinjlim B_i$ is a direct limit of the system $\{B_i\}$.
I won't prove that it's a functor, but to get an idea, suppose that you have two systems $\{B_i\}$ and $\{C_i\}$ and a collection of morphisms $h_i : B_i \to C_i$, compatible with the systems. Then you can compose with the "injection" morphisms to get $\tilde{h_i} : B_i \to C_i \to {\varinjlim}_j C_j$, and these maps commute with the maps of the system $\{B_i\}$. So by the universal property you get a map $$\varinjlim h_i : \varinjlim B_i \to \varinjlim C_i.$$
It remains to show that $\varinjlim (h_i \circ g_i) = \varinjlim h_i \circ \varinjlim g_i$, a healthy exercise.
