Why do we need the axiom MS3 for localizing categories? 
Background: Let $\mathsf{C}$ be a category and $S$ be a collection of morphisms (let's suppose that $S$ has all the identities and is closed under multiplication just to simplify a bit). We can construct the localization $S^{-1}\mathsf{C}$ as a category such that $\operatorname{Ob}(S^{-1}\mathsf{C})=\operatorname{Ob}(\mathsf{C})$ and such that a morphism is an equivalence class of strings of the form $s_0^{-1}\circ f_1\circ s_2^{-1}\circ f_3\circ \cdots\circ f_n$, where we impose all the obvious compatibility conditions such as $s^{-1}\circ s\sim\operatorname{id}$. (These compatibility conditions can be found on the page 2 of Dragan Milicic notes.)

One problem with localization in all its generality (as described above) is that if $\mathsf{C}$ is an additive category, it is not clear if $S^{-1}\mathsf{C}$ is also additive or not. For that, we impose the axiom "LMS2":

This implies that every morphism in the localization can be represented as $s^{-1}\circ f$ and that any finite number of morphisms can be represented using the same "denominator" $s$. In particular, we can define an additive structure on $S^{-1}\mathsf{C}$ as
$$s^{-1}\circ f + s^{-1}\circ g=s^{-1}\circ (f+g).$$
In basically every reference that I could find, the authors also required an axiom "LMS3" given by

I feel like this last axiom is only useful for giving a simpler (is it really simpler?) description of the equivalence relation between morphisms without needing to ever consider strings of length larger than 2.
Is there a real need for this axiom LMS3 that I didn't see?
 A: The axiom LMS3 you ask about directly corresponds to the axiom that any parallel pair of arrows in a filtered category can be coequalised.
The idea is to consider the full subcategory of $\mathcal{C}_{/ N}$ spanned by the $(L', s')$ where $s' : L' \to N$ is in $\mathcal{S}$, which by a minor abuse of notation I will denote by $\mathcal{S}_{/ N}$.
(If $\mathcal{S}$ has the 2-out-of-3 property and contains all identity morphisms then this is no abuse of notation.)
Given an object $(L', s')$ in $\mathcal{S}_{/ N}$ and a morphism $h : L' \to P$ in $\mathcal{C}$, we have a morphism $h \circ (s')^{-1} : N \to P$ in $\mathcal{C} [\mathcal{S}^{-1}]$.
This defines a cocone from the obvious diagram of shape $(\mathcal{S}_{/ N})^\textrm{op}$ to $\mathcal{C} [\mathcal{S}^{-1}] (N, P)$, and hence a map
$$\varinjlim_{(L', s') : (\mathcal{S}_{/ N})^\textrm{op}} \mathcal{C} (L', P) \to \mathcal{C} [\mathcal{S}^{-1}] (N, P)$$
When you have a Gabriel–Zisman calculus of fractions, $(\mathcal{S}_{/ N})^\textrm{op}$ is filtered and the comparison map is a bijection.
Filtered colimits have many convenient properties and in this situation we are able to easily transfer convenient properties of $\mathcal{C}$ – e.g. existence of finite limits and/or colimits – to $\mathcal{C} [\mathcal{S}^{-1}]$.
