What does multiplying two real-world values represent? I totally get what division means in the real world.  "dollars / hour", well, that's the number of dollars you will make in one hour.  "kilometers / gallon" is the distance you can go with a gallon of gas.  Division means that given a certain amount of one thing, you'll get a certain amount of another thing.
I'm so good with division that you could give me a ratio I've never seen before and I can tell you what it means.  "Burgers / McDonalds"?  It's the average number of burgers a McDonalds will produce.  "Dolphin / Miles"?  Every mile you drive, you get this amount of dolphins.
Multiplication on the other hand makes no sense at all.  What the heck is a foot-pound?  And I don't really mean the definition.  I mean, what does multiplication actually do for these two values?  What does multiplying two values say about their relationship?  What if I were to say kilometer-hours? Or dollar-kilograms? Or dolphin-miles?  What would those things mean?  
Bonus points go to explaining this in a very simple, clear manner.
 A: One way to think of different dimensions multiplied together is as a weighting factor that helps transform your unit of measure into something else.  I know that this doesn't sound that helpful, but think of the following.  
A foot-pound is a unit of torque.  It is a measure of 1 pound of force, applied 1 foot away from a pivot point.  The distance is a weighting factor that transforms the force you apply into a torque.  Larger/smaller distance transform your applied force into larger/smaller torques.  
Similar things can be said for other units, such as Newton-meters, which transforms a Force (Newtons) into an energy by weighting it by the distance over which the force is applied.  (I realize now that this is the same units as above, but for a different quantity, energy vs torque)
As for some of the more strange units you've listed above, for instance, dolphin-miles, you could use this as a measure of how far dolphins a certain number of dolphins are from a certain point.  Adding many dolphin-mile quantities together, and dividing by the total number of dolphins gives you the average position of the dolphins.
You could also use this as a measure of the total distance traveled by a group of dolphins.  If 10 dolphins each travel 10 miles, then you would have 100 dolphin-miles of travel.  (the same goes for Frisbee's man-hours comment above, which is what reminded me to put this)
Admittedly, things do get weird, because the final unit has to be something that you can make sense of, but this is one way to think of it.
A: Man-hours is nonsensical to you? 2 men worked on it for 5 hours is 10 man-hours.  
Momentum is mass X velocity   
kilowatt hour
kinetic energy is 1/2 m X v X v  
Multiplication of two real world values represents exactly that 
Kilometer / McDonald makes sense?
A: The main point here is that any division is indeed a multiplication. For example, let's use your example of the Burgers $(b)$ and McDonalds $(m)$:
\begin{equation}
 \frac{b_{bu}}{m_{mc}}=b_{bu}\times\frac{1}{m_{mc}}=a_{bu/mc}
\end{equation}
where $a$ is the average number of Burgers per McDonalds' stores. In this case, what make sense is to multiply the number McDonals's stores with the ratio that was calculated, it is:
\begin{equation}
 m_{mc}\times a_{bu/mc}=b_{bu}
\end{equation}
which means that if you multiply the number of McDonalds' stores with their average production of burgers, you will obtain the whole production of burgers.
In your question, the following multiplication:
\begin{equation}
 m_{mc}\times b_{bu}=c_{mc\times bu}
\end{equation}
does not make any sense because the units of the product's result $(mc\times bu)$ does not make any sense in this context. If you were to interpret this result, you would be saying that any McDonald store produces a $b$ amount of burgers (every store produce the overall production!!). That's why you cannot make sense of the product that you stated.
The lesson here is that when doing any kind of math operation, you should never forget about the units of measure of the factors involved as well as the unit of measure of the result. One of the best examples in physics is provided for the gravity acceleration constant $(g)$:
\begin{equation}
   g\approx 9.80_{\frac{m/s}{s}}
\end{equation}
which means that any falling object in the earth falls in a speed of 9.80 meters per second, but such speed also increases every second (or, in other words, it accelerate while falling).
