First of all: French effacer means "to erase", and it is of course a composite of the prefix e- (latin ex-) [away from] face. I'm not aware of a different meaning than erase (maybe wipe up or annihilate are also viable translations in some circumstances), and what you're describing is probably more or less what Grothendieck had in mind. As far as I know Grothendieck is the inventor of this terminology.
Let me focus on the simplest case:
Let us assume that the abelian category has enough injectives and let $F$ be a (left exact) functor to another abelian category. As you know, the associated right derived functors $R^nF$ can be constructed by choosing an injective resolution of each object, applying $F$ to the resolution and taking homology.
The right derived functors are a universal $\delta$-functor due to the fact that each $A$ admits an injection $A \to I$ such that $R^{n}F(I) = 0$ for $n \geq 1$.
Let us look at a specific instance of the arguments involved: The first step in proving that given any other $\delta$-functor $T^n$ and a natural transformation $\phi : F \Rightarrow T^0$ there is a unique sequence of natural transformations $\phi^n: R^nF \Rightarrow T^n$ extending $\phi$ and compatible with the connecting morphisms of both $R^nF$ and $T^n$.
This sequence of natural transformations is built by induction. For each $A$ choose an injective effacement $0 \to A \to I \to C \to 0$ and apply both $R^nF$ and $T^n$ and the induction hypotheses. We get to the following diagram:
$\begin{array}{ccccccc} R^nF(I) & \to & R^{n}F(C) & \to & R^{n+1}F(A) & \to & \mathbf{0} \\ \downarrow & & \downarrow & & \downarrow \exists? & & \\ T^n(I) & \to & T^n(C) & \to & T^{n+1}(A) \end{array}$ The bold zero is due to effaceability of $R^{n+1}F$. The two left hand vertical maps are already constructed and the map marked with a question mark exists and is unique by an easy diagram chase because the bold zero implies that the connecting morphism $R^{n}F(C) \to R^{n+1}F(A)$ is an epimorphism. The map just constructed is the $A$-component $\phi_{A}^{n+1}$ of $\phi^{n+1}$. Such arguments are applied some further times when checking that this construction actually defines a natural transformation $\phi^{n+1} : R^{n+1} F \Rightarrow T^{n+1}$, that the $\phi^n$'s are compatible with the connecting morphisms and that they are unique in that they don't depend on the chosen effacement.
In summary and conclusion:
Universality follows from effaceability due to the fact that the connecting morphisms $R^{n}F(C) \to R^{n+1}F(A)$ associated to the sequence $0 \to A \to I \to C \to 0$ are forced to be epimorphisms, as $R^{n+1}F(I) = 0$. In other words, because $R^{n+1}F$ gets erased at $I$.
I hope that this is clear enough, if not, I strongly recommend to go through the proof of universality of derived functors once again, e.g. in Theorem 2.4.7 of Weibel. (Don't miss the exercises at the end of the section!)
The effaceability condition can be used to show more generally: If $S^n$ is a $\delta$-functor and $S^{n}$ is effaceable for $n \geq 1$, then the $\delta$-functor $S^n$ is universal.
This is used when one applies dimension shifting: Suppose we have an effacement $0 \to A \to I \to C \to 0$ and $S^{n}(I) = 0 = S^{n+1}(I)$. Then the sequence $0 \to S^n(C) \to S^{n+1}(A) \to 0$ shows that the connecting morphism $S^n(C) \to S^{n+1}(A)$ is an isomorphism because of this effacement.
For instance, the definition of flatness implies that $Tor_{n}(A,F)$ is zero for a flat module $F$ and $n \geq 1$. A variant of the previous argument can be used to show that $Tor_{n+1}(A,B)$ can be constructed from $Tor_{n}(A,-)$ using a flat (co-)effacement of $B$, and the resulting $\delta$-functor is universal due to the above.
Sometimes a slightly more general variant of the effaceability is called Buchsbaum's criterion (or weak effaceability) due to Proposition 4.2 in Buchsbaum's Annals paper Satellites and Universal Functors.
Added:
Here is, to the best of my knowledge, the most useful general criterion for $F$ to admit a (classical) right derived functor, which I learned from Bernhard Keller. It covers all the "oodles of applications" explained by Akhil in his answer:
Suppose $\mathcal{C}$ is an abelian (or more generally an exact) category and assume that $F:\mathcal{C} \to \mathcal{D}$ is a functor to an abelian category $\mathcal{D}$.
Theorem. If there exists a full subcategory $\mathcal{A}$ of $\mathcal{C}$ such that
- $\mathcal{A}$ is closed under extensions: if 0 \to A' \to C \to A'' \to 0 is exact in $\mathcal{C}$ then $C$ is an object of $\mathcal{A}$.
- The restriction of $F$ to $\mathcal{A}$ is exact.
- For every object of $\mathcal{C}$ there exists a monomorphism $C \to A$ with $A$ in $\mathcal{A}$.
- For every short exact sequence 0 \to A' \to C \to C'' with A' in $\mathcal{A}$ there exists a commutative diagram \begin{array}{ccccccccc} 0 & \to & A' & \to & C & \to & C'' & \to & 0 \\ & & \parallel & & \downarrow & & \downarrow \\ 0 & \to & A' & \to & A & \to & A'' & \to & 0 \end{array} where the second row is exact and A, A'' are in $\mathcal{A}$.
Then $F$ admits a right derived functor.
Notes:
- The technical condition 4. is satisfied in presence of 3. if $\mathcal{A}$ is supposed to be closed under quotients in $\mathcal{C}$. That is: Whenever 0 \to A' \to A \to C \to 0 is short exact then $C$ is an object of $\mathcal{A}$.
- No left exactness of $F$ on $\mathcal{C}$ is imposed: this is only needed to guarantee that $R^0F = F$. In general, $R^{0}F$ is the best left exact approximation to $F$. In highbrow terms: $R^{0}$ can be seen as reflection of the inclusion of the left exact functors into all additive functors.
- The right derived functors of $F$ can be computed by choosing for each $C$ an $\mathcal{A}$-resolution — which exists due to the effaceability conditions 2 and 3.
- The objects of $\mathcal{A}$ are called $F$-acyclic or adapted to $F$ if $\mathcal{A}$ satisfies the hypotheses of the theorem.
Final remarks (coming back to effaceability):
The above theorem admits a partial converse: if $F$ admits a right derived functor, the category $\mathcal{A}(F)$ of $F$-acyclic objects (a notion I don't want to define in full generality here), satisfies conditions 1., 2. and 4. above. The effaceability of the higher right derived functors is thus seen as the part missing from a complete characterization of derived functors.
The usefulness of injective effacements is of course that injective objects are $F$-acyclic for every additive functor. This is because short exact sequences of injectives split, by definition of injectivity. In particular, every functor on an abelian category with enough injectives can be derived (not only left exact ones).
I'm not aware of a complete proof of the theorem as stated in the literature without making a detour via derived categories, which certainly looks like overkill given its rather elementary nature. See sections 12ff in Keller's article or sections 10.5, 10.6 and 12 in my notes on exact categories.