What exactly is a kilogram-meter?

The kilogram-metre (or more often the kilogram-kilometre) is a unit of measurement indicating that 1 kg (lets assume of coal) has been moved 1 km (towards, for example, a power station). It's used by big freight companies, governments etc. as one metric of how much hauling they're doing. As you can imagine they're often they're very concerned with CO$_{2}$ emissions per kilogram-metre.

Wikipedia link here.

(I realise you asked for an explanation of general multiplied units, rather than about the kg-m specifically, but there are already a few general answers).

By your question you could see divided units as rate, an amount of one quantity would be changed based on the amount of another. When looking it at that way you could think about multiplied units as conserved quantities, if you would double one you should half the other to have the same effect. Some interesting examples could be:

  • Power $P = U I$ thus in volt ampere;
  • Torque $\tau = F r$ thus in Newton meters (rotating a door).

Combining these two idea's about divided and multiplied units could be done by division on one of the sides, taking the identity $U = RI$ or in units $$ \mathrm{V} = \Omega \mathrm{A} \Leftrightarrow \Omega = \frac{\mathrm{V}}{\mathrm{A}}.$$ Where you could take two views. From the point of division the resistance is the amount of power required at a certain current. Or from the point of multiplication, you could keep the same voltage difference by increasing the value of the resistor by the same factor as decreasing the current.

Multiplied units often turn up in "double proportion" situations, where a particular quantity is proportional to multiple dependent variables. To take a deliberately non-physics example, a difficult task might need either many people to work on it or people to work on it for a long time, so its difficulty is proportional to both the number of workers and the time spent and could thus be measured in "person-days". A six person-day job would require two people to work for three days to complete it, or three for two days, or whatever. (The validity or otherwise of the model in which these two working schemes are equivalent is left as an exercise for the reader!)

Moving to a fairly exact parallel from a physics context, an electron-volt - or, to be pedantic, an electron-charge-volt - is a good unit of energy because the work done to move a charge through a potential difference is proportional to both the charge and the potential difference. The same goes for a coulomb-volt, although of course we just call that a joule.

Sure, you can explain this by cancelling out SI base units, but to some extent that's a red herring. If we defined the coulomb and the volt as base units, we would be perfectly happy to define our unit of energy as a coulomb volt.