Einstein summation convention dictates that repeated indices should be summed. Thus the equation $a_{ij} = b_{ik}c_{kj}$ is taken to mean $a_{ij} = \sum_k b_{ik}c_{kj}$ where in both cases the range of summation is implicit.
Oftimes when I have come across this notation, it is followed by the statement "where summation over index $k$ is implicit." This seems to defeat the point of Einstein notation (to reduce clutter in equations).
Othertimes, the summation is not obvious (as it may be above). For example, if asked to evaluate $F^{\mu \nu}F_{\mu \nu}$, one might think that the answer depends on the values of $\mu$ and $\nu$, but in actual fact, summation is implied.
Given these ambiguities and failure to reduce clutter (well, rather trading clutter in equations to clutter in text), why should one use Einstein notation?
Einstein notation is a coordinate-based implementation of abstract index notation when there is a fixed set of bases for all vector spaces. This is the same as how matrices are coordinate-based implementations of linear maps between vector spaces.
In turn, abstract index notation is a highly convenient notation for chaining together complex combinations of multilinear functions, and fluidly converting inputs of multilinear functions to outputs and vice versa.
The inputs of a multilinear function can be turned into outputs, and vice versa, by viewing certain inputs to the function as "fixed" and other inputs as "free". Everytime you convert an input to an output or vice versa, you dualize the relevant space. For example, consider a multilinear function with 3 inputs: $$T(\cdot, \cdot, \cdot) : U \times V \times W \rightarrow \mathbb{R}.$$ It can instead be viewed as the following multilinear function with the first two spaces as inputs, and the dual of the third space as the output: \begin{align} &\tilde T(u,v) \mapsto T(u,v,\cdot), \\ &\tilde T :U \times V \rightarrow W^* \end{align}
Given a collection of multilinear functions, the outputs of one can be plugged in as inputs into another whenever the spaces are compatible, creating an interconnected network of multilinear functions.
Continuing the example, say $S$ is another multilinear function with $W^*$ as an input space: $$S:X \times W^* \mapsto \mathbb{R}.$$ Then one can define a new multilinear function composing the two: $$(x,u,v) \mapsto S(x,\tilde{T}(u,v)).$$
Even though this combination of multilinear functions is conceptually simple, it was clunky to write down (requiring the definition of an auxiliary function $\tilde{T}$). Instead, one can use abstract index notation to write the same thing concisely as: $$T_{uvw}S_x^w.$$