As Peter Suber (Philosophy Department, Earlham College) points out in his website titled “Paradoxes of Material Implication”, material implication is the price of truth-functionality. Here is the link:
These are paradoxes in the ancient sense, violations of intuition. They are not contradictions. But, you may well ask, why would we adopt a type of implication with such counter-intuitive results? … Primarily, the answer is that we want a truth-functional kind of implication. Remember that a connective is truth-functional if we can figure out the truth-value of the statement solely on the basis of the truth-values of its components. If we use a truth-functional form of implications, then we can construct truth-tables for our implication statements
Edit: I forgot to point out that material implication is not unknown in ordinary language. When I was about 6 years old, trying to catch one of the birds that lighted in our yard, my grandfather gave me some this joking advice: “If you want to catch a sparrow, all you have to do is put salt on its tail.” :-)
Further edit: Nonetheless, there is a tendency to avoid material implication. A good example is the Wikipedia article on absolute value, at the line:
If b > 0, two other useful properties concerning inequalities are:
(We need to digress a moment to point out that this line ought to be edited to say that b is non-negative. That is, the case where b = 0, although trivial, is still germane. In our discussion below, we will assume that such an edit has been done.)
By material implication, it is irrelevant if b is non-negative, but, admittedly, relevance disappears if b < 0. Perhaps the chin-strokers at Wikipedia feel that they are making it easier to understand for the layman by filtering the “irrelevance”, but a heavy price is paid for that, namely, the disruption of the nice chart that they were creating. If you simply let material implication do its work, then you can continue with the chart in an uninterrupted way.
Further edit: Another example of the avoidance of material implication is in the definition of a “critical point” of a function, namely, the usual practice of saying that a point c is a critical point of a function f if, and only if, c is in the domain of f, and either f’(c) = 0 or f’(c) does not exist. A niftier way of putting it is that c is in the domain of f, and f is differentiable at c implies f’(c) = 0. Period. End of story. Anyone who asks “What if f is not differentiable at c? Is c a critical point or not in that case?” labels themself as mathematically, or at least logically, illiterate. (I hope I don’t get labelled as being literarily illiterate for using “themself”! - see the discussion in Wiktionary.) Of course, if the student asks this, then the teacher must answer, and the answer is: “Yes, by material implication. That is, if f is not differentiable at c, then, by material implication, the implication if true, and so c is a critical point of f in that case as well.” The formulation using material implication is niftier not only for being more concise, but also for avoiding explicit reference to something non-existent. Knowledge of and acceptance of material implication should be established and assumed at an early point in one’s mathematical training. Notice that, unlike the example in Wikipedia regarding absolute value, material implication in this case addresses a condition that is relevant.
I daresay that this example of spoon-feeding filtering of irrelevance is quite widespread, and may even be partly contributory to the difficulty in appreciating material implication when those moments come when it cannot be avoided. That is, if the unrequested filtering were not routinely done, then people would be used to material implication already, just as they are used to tricky idioms in ordinary language, complex word-play jokes, and so on. In other words, it is the needless UNFAMILIARITY with material implication that is partly to blame here.
Material implication often involves the null set. For example, the intersection of the null set is, by material implication, the universal set. Letting material implication work for you this way means that you do not have to separately consider the null case. Including the null case, by material implication, into the set of all cases allows for unity of treatment, and recognition of otherwise hidden patterns. For example, a selfie is basically a photobomb in which the victim is non-existent (i.e., null). (A photobomb might be accidental, or involve something other than the image of the perpetrator, and so the precise definition of a selfie would be “an intentional victimless photobomb featuring the image of the perpetrator”).
It is because of material implication that a minute of silence (such as at the beginning of a school day) is not in violation of the "English only" rule that may be in effect.
A classic example of material implication is one known from antiquity, namely, Archimedes' boast about the lever: "Give me a place to stand, and I will move the Earth." - which is just an emphatic way of saying, "If I had a place to stand, I could move the Earth."
Material implication is also used by religion to raise the topic of transcendence, transcendence being the raison d'être of religion. For this reason, the false assertions of religion are not a defect of religion, but an essential part of its dynamic. For example, is it not true that if an infinite amount of time has passed, then 2 plus 2 equals 5? By material implication, it is true. The assertion that 2 plus 2 is 5 is not really of interest. It is like the nail in the story about nail soup. What is of interest is that the subject of "an infinite amount of time" - in other words, transcendence, has been raised.
further edit: A nice (“classic”?) example of material implication at work is the fact that symmetry together with transitivity does not imply reflexivity. This situation is dealt with in detail on page 30 of the book ESSENTIALS OF ABSTRACT ALGEBRA by Bundrick and Leeson (1972), and is listed as one of the answers here in MSE to the question about “obvious” theorems that are actually false, at the following location:
'Obvious' theorems that are actually false