When I teach the MVT in a Calculus class, I do three things:

a) Show the one real-world example I know and which everyone gets: Police has two radar controls at a highway, say at kilometre $11$ and at kilometre $20$. Speed limit is $70$ km/h. They measure a truck going through the first control, at 11.11am, at $65$ km/h, and going through the second control at 11.17am, at $67$ km/h. They issue a speeding ticket. Why?

Let the class think about this. Every time I've taught this, someone realised after a short while that the truck passed $9$ km in $6$ minutes, so its average speed was $90$ km/h. Then someone says something like: **you cannot go at an average speed of $90$ km/h without ever going at a speed of $90$ km/h** (and certainly not without ever going more than $70$ km/h). This is totally common sense, but also it is exactly the MVT. Draw a graph of the position function, realise that the numbers $65$ and $67$ were just red herrings (tangent slopes at the endpoints, irrelevant to the argument), discuss whether there is some way out: Can the function have discontinuites? Well, a jump discontinuity would be a wormhole the truck fell through, or more realistically some shortcut off-highway which would be illegal too. Points where the derivative does not exist? ~~Actually yes, if the truck braked somewhere, but it cannot have done that more than finitely many times, and then we break down the problem into subintervals.~~ Turns out: No, even sharp braking cannot create a "sharp turn" of the function under standard assumptions of physics, see comments by users @leftaroundabout and @llama.

b) Mention that aside from that, it is a "workhorse theorem" which we never see but which **makes the entire curve sketching routine work**. How do you prove **Positive derivative means increasing function**: with the MVT. How do you prove **Derivative $0$ on an interval means constant**: with the MVT. Of course we never think of the proofs of those, we just use them as "well-known", but without MVT, they would not be there.

c) Related to b, it comes up crucially in the **Fundamental Theorem** later, compare Arturo Magidin's answer. I point it out when I'm there.

**Added**: As this answer seems to get a lot of attention, I want to put in one more thing part of which I try to get across in class when the MVT is up.

d) The derivative is a cool thing because it carries a lot of information about the original function, but in a subtle way. To the non-initiated, the graphs of an $f$ and $f'$ would most often look totally unrelated. But the initiated, i.e. your calculus class, at this point should already "get" intuitively "hmm, $f'$ is very negative around here, so $f$ should decrease with a steep slope in this neighbourhood". Now the MVT is *the one theorem which attaches actual numbers to this intuition*, it is **the first result which gives an explicit (albeit subtle) relation between values of $f$ and values of $f'$**. That is why it underlies the proofs of all the fancy machinery that, later, gives seemingly much stronger relations between $f$ and $f'$, like Curve Sketching, the Fundamental Theorem, Taylor Series, and even L'Hôpital's rules (thanks @JavaMan for pointing out this one). They get all the limelight, but in a way, they all are refined versions of repeated applications of the MVT plus special conditions.

Further update: Since the "speeding" application of the MVT keeps getting mentioned everywhere (and of course I don't even remember where I got it from originally), I googled a little and see that it's been around for quite a while. This educational video of the MAA's from 1966 is almost of historical value (although I hardly understand the voice-over due to its *very American* accent). As for the question whether this is actually done, thanks to User Bracco23 for providing one source from Italy in a comment. Here is another one from Scotland: http://news.bbc.co.uk/2/hi/uk_news/scotland/4681507.stm The internet has more hearsay and debates: 1 2 3. Cf. also this answer.