What are some physical, geometric, or otherwise useful interpretations of divergent series? I don't understand what ideas such as Abel, Cesàro summation or other types of sum 'regularization' help us describe. What is the practical application to discussing the 'sum' of sequences that are not convergent in the usual sense? 
What is the motivation to assigning a value to an otherwise divergent sequence, and further why is it a good idea to call whatever comes out a 'summation'?
 A: Series that are classically divergent, and even Cesàro divergent, play an important role in physics. 
The canonical example of this is the Casimir effect. When calculating the force of the effect, you are confronted with a divergent series. This series diverges even when you attempt to Cesáro sum it. However, if you use a technique known as zeta function regularization, you can recover a finite and physically meaningful quantity.
A: There is a lot of hype around about "values" of certain divergent series, like $\sum_{k=1}^\infty (-1)^kk$. But so far nobody has come up with a way of assigning a value to such a series which could be called "universal", or "canonical". 
It is, however, true that in some cases a divergent series $\sum_{k=1}^\infty a_k$ can be considered as a limiting case of a family of convergent series $$s(x):=\sum_{k=1}^\infty a_k(x)\qquad(x\in U)\ ,\tag{1}$$
whereby at the same time for all $k$ the limits
$$\lim_{x\to\xi} a_k(x)=a_k$$
exist, as well as the limit $\lim_{x\to\xi} s(x)=:\sigma$. One is then tempted to say that $\sum_{k=1}^\infty a_k=\sigma$, but making this conclusion is in fact voodoo mathematics.
Note that the "embedding" of the given series $\sum_{k=1}^\infty a_k$ into an environment $(1)$ involves many arbitrary choices by the master of ceremonies.
A: Indeed. 
When studying Fourier series, the summation method that is ideal to get uniform convergence of the series to the function is Cesaro summation. See Fejer's theorem.
There are functions that are analytic, for which the Taylor series converges to the function. But there are functions for which this is not true. Using linear summation methods, like Cesaro summation you get the function back. This is a theorem of Carleman that all functions in a quasianalytic Denjoy-Carleman class can be recovered in this way, for certain linear summation method (not only Cesaro's).
You may want to look also to a nice theorem describing all linear summation methods (consistent with the usual summation) by Silverman and Toeplitz.
There are non-linear summation methods too. For example Shanks summation and its generalizations using Pade' approximants. 
Notice that the way we assign a sum to a series is, to begin with, already kind of an arbitrary choice. Then, why not consider others.
A: It is similar as to ask for an ancient greek mathematician: "what is the practical reason to consider so called 'irrational numbers', numbers like $c$ in $c^2 = 1^2+1^2$ , which are in no measurable relation to the whole numbers and their fractions?"           
Or of a medieval european: "what is the practical reason to try to derive roots of $-1$ where everyone knows there is no number which multiplied with itself is $-1$?" 
Or, now coming to divergent series: "Well, we can denote rational numbers by the fraction $f(x)= {1 \over 1 -x}$, for instance,$f(0)=1$ or $f(1/2)= 2$ or $f(2)=-1$ except, of course $f(1)$ which gives $1/0$ which is undefined. And, of course we can do the long polynomial division with $f(x)= {1\over 1-x}=1+x+x^2+x^3+...$, which is never ending and just appends one more power of $x$ for one more step of expanding. But what is the practical reason to set $x=2$ into that power series and assign $-1$ as its value?"           
And so on. See lots of series, important series even, which gives finite values for some continuous range of argument and infinite for some other continuous range, like in the example before. And which simply occur by "long-division" in some cases. The powerseries for $\sqrt{1+x}$ gives for lots of $x$ divergent series - but why shouldn't we try to assign the expected values at them if that does not confliges with other results (which, however, is of course the critical condition).        
So this might be seen as an axiom that divergent series might in specific cases get a finite value assigned, derived by some standard procedere which can be finally regressed to some most basic standard procedures like 'Abel' summation (derived from the idea for the geometric series), 'Zeta'-regularization and only few more. If that axiom and its derivations do not conflige with standard math in the convergent cases, we can build a technology around it. See zeta-regularization and that this regularization even agrees with the physical world when we model it with that regularized results. (see for instance the "Casimir-effect" mentioned in the other answer)
