In full generality, derived functors appear in a situation where our categories of interest admit some notion of weak equivalences, without those being isomorphisms per se. In homological algebra, our cochain complex categories have quasi-isomorphisms, which are such a notion of weak equivalences. These quasi-isomorphisms are interesting because they are (by definition) maps that preserve the cohomology of the complex, so in case we study cochain complexes only for the purpose of cohomology (which is what homological algebra does), this is actually the natural notion of equivalence. The traditional (1-categorical) problem is of course that they are not invertible and not isomorphisms in the category of cochain complexes.
This means that a general functor $F\colon\mathrm{Ch}^{\geq 0}(\mathcal{A})\to\mathrm{Ch}^{\geq 0}(\mathcal{B})$ between bounded cochain complex categories need not preserve quasi-isomorphisms. In homological algebra (which is all about cohomology), the natural question is therefore if there is some universal way to turn $F$ into a functor that does preserve quasi-isomorphisms. If we are slightly more precise what we mean with ''universal'', then such a universal way to turn $F$ into a functor that does preserve quasi-isomorphisms gives us the derived functor of $F$. In general, if $F\colon\mathcal{C}\to\mathcal{D}$ is a functor between categories with weak equivalences, then a derived functor for $F$ is a functor that preserves weak equivalences and is in a specific universal way obtained from $F$. I will say later more about what universality specifically means here.
Let $F\colon\mathcal{A}\to\mathcal{B}$ be a left exact functor. We will write $F_*\colon\mathrm{Ch}^{\geq 0}(\mathcal{A})\to\mathrm{Ch}^{\geq 0}(\mathcal{B})$ for the induced functor on cochain complexes. Even though $F_*$ might not preserve quasi-isomorphisms, it does preserve cochain homotopy equivalences. The next step is best understood in the context of model categories.
A model category $\mathcal{M}$ is informally a complete and cocomplete category with a notion of weak equivalences, and a notion of cofibrations and fibrations, that heavily control the behaviour of the weak equivalences (making it possible to actually understand those weak equivalences, which are the main things we care about). It is important to mention that weak equivalences must satisfy the 2-out-of-3 property: if $f$ and $g$ are composable morphisms, then if two out of the morphisms $f$, $g$ and $gf$ is a weak equivalence, so is the third.
Model categories also admit a notion of homotopy equivalences, which are a special kind of weak equivalences. An object $Y$ of a model category $\mathcal{M}$ is fibrant if the unique map $Y\to 1_\mathcal{M}$ to the terminal object of $\mathcal{M}$ is a fibration. Dually, and object $X$ of $\mathcal{M}$ is cofibrant if the unique map $\varnothing\to X$ from the initial object of $\mathcal{M}$ is a cofibration. An object is bifibrant if it is both fibrant and cofibrant. Any model category $\mathcal{M}$ comes with a functor $B\colon\mathcal{M}\to\mathcal{M}$ that sends an object $X$ to a bifibrant object $BX$ equipped with a natural weak equivalence $X\simeq BX$. $B$ is called a bifibrant replacement functor.
The crucial thing is that it turns out that, in any model category $\mathcal{M}$, a morphism $X\to Y$ between bifibrant objects is a weak equivalence if and only if it is a homotopy equivalence. (This is a generalization of the Whitehead theorem that weak homotopy equivalences between CW-complexes are homotopy equivalences, because CW-complexes are bifibrant in the Quillen model structure on topological spaces.) Therefore, if $F\colon\mathcal{M}\to\mathcal{N}$ is a functor that preserves homotopy equivalences, we can look at the functor $FB\colon\mathcal{M}\to\mathcal{N}$, where $B$ is bifibrant replacement. This functor $FB$ will preserve all weak equivalences: if $X\to Y$ is a weak equivalence, then $BX\simeq X\to Y\simeq BY$ gives us the induced map $BX\to BY$, and this is a weak equivalence by the 2-out-of-3 property of weak equivalences. But a weak equivalence between the bifibrant objects $BX$ and $BY$ is a homotopy equivalence, and $F$ preserves those. The functor $FB\colon\mathcal{M}\to\mathcal{N}$ is the derived functor of $F$.
Since you are looking at a left exact functor $F\colon\mathcal{A}\to\mathcal{B}$ be a left exact functor, which induces a cochain homotopy equivalence preserving functor $F_*\colon\mathrm{Ch}^{\geq 0}(\mathcal{A})\to\mathrm{Ch}^{\geq 0}(\mathcal{B})$, in order to find a derived functor of $F_*$ it suffices to find a model structure on $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ in which weak equivalences are quasi-isomorphisms, and homotopy equivalences are cochain homotopy equivalences. One model structure that does this is the injective model structure on cochain complexes, so we have found our way to define the derived functor of $F_*$.
It turns out that every object in the injective model structure is cofibrant. Moreover, the fibrant objects are precisely the cochain complexes in which all terms are injective in $\mathcal{A}$ (hence the name). A bifibrant replacement functor $B$ will hence assign to a cochain complex a quasi-isomorphic cochain complex (as our weak equivalences are quasi-isomorphisms) in which are terms are injective. This is precisely an injective resolution. So, the derived functor of $F_*$ can be given as follows: take a cochain complex, pick an injective resolution, and then apply $F_*$ to this injective resolution. This is where the standard recipe you learn in homological algebra comes from. It is now standard to define the derived functors $\mathbf{R}^i F$ to be the composite $A\mapsto\mathrm{H}^i\circ\mathbf{R}F_*(A[0])$.
To summarize: you want functors that do not preserve weak equivalences to become functors that do, in some universal way. In homological algebra, the presence of a model structure allows us to do so.
This leaves one question: what do I mean with ''universal''? I do not see a simple way to explain this, so when I get precise below, it may be a bit abstract. The simplest I can put it is this: a derived functor should be the homotopical functor that is, among all other homotopical functors, the ''closest'' approximation of the original functor.
Now, you will notice that, in principle, the bifibrant replacement functor $B$ in a model category could be replaced by any other such functor $B'$ for which $BX\simeq B'X$ naturally for all objects, and we could define the derived functors using $B'$. So the derived functor is not unique up to isomorphism, at most it is unique up to weak equivalence. This is in some sense not surprising, since derived functors are a purely homotopical notion. I wrote above that a traditional problem of weak equivalences is that they are not invertible in the 1-category in which they live. Given a category $\mathcal{C}$ with a class of weak equivalences $\mathcal{W}$, there are two solutions to this: either, we formally invert all weak equivalences and pass to the localization $\mathrm{Ho}\,\mathcal{C}:=\mathcal{C}[\mathcal{W}^{-1}]$, or we build a so-called $\infty$-category $\mathcal{C}_\infty$ out of this data, which is a higher-categorical structure that allows for weak equivalences to be invertible in a higher homotopical sense. We can build $\mathcal{C}_\infty$ as $\infty$-categorical localization of the $1$-category $\mathcal{C}$ at the class of morphisms $\mathcal{W}$ (this differs from the $1$-categorical localization).
Write $h\mathcal{C}$ for either $\mathrm{Ho}\,\mathcal{C}$ or $\mathcal{C}_\infty$. The former is called the homotopy category of $\mathcal{C}$, and both are in algebraic settings known as the derived ($\infty$-)category of $\mathcal{C}$. Given any functor $F\colon \mathcal{C}\to\mathcal{D}$, where $\mathcal{D}$ also admits a notion of weak equivalences, a total left derived functor of $F$ is a(n $\infty$-)functor $\mathbf{L}F\colon h\mathcal{C}\to h\mathcal{D}$ such that the (possibly non-commutative) diagram
$$
\require{AMScd}
\begin{CD}
\mathcal{C} @>{F}>> \mathcal{D}\\
@V{\mathrm{localization}}VV @VV{\mathrm{localization}}V\\
h\mathcal{C} @>{\mathbf{L}F}>>h\mathcal{D}
\end{CD}
$$
witnesses $\mathbf{L}F$ as a right Kan extension. A functor $F'\colon\mathcal{C}\to\mathcal{D}$ is a left derived functor of $F$ if it preserves weak equivalences, comes with a natural transformation $\eta\colon F'\to F$, and if this natural transformation witnesses the induced functor $hF'\colon h\mathcal{C}\to h\mathcal{D}$ (this induced functor exists by the universal property of localization) as a total left derived functor for $F$. We can dually define total right derived functors and right derived functors.
