Are they familiar with traditional burials? Digging a hole in the ground and interring the body (covered in wrappings or housed in a coffin of some sort) and marking the grave with some sort of headstone? If they are, then you could say that cremation is the most space-efficient way to dispose of mortal remains (it's a big planet, but with too many people to be able to bury everyone everywhere, there are alternatives). The full-sized body is reduced to ashes (in a special furnace that gets super duper hot--you don't really want them looking funny at your kitchen oven); the ashes take up a lot less space than an entombed body. You still have to pay for the cremation and some sort of holding vessel for the ashes, but you don't have to find a cemetery, pay for the excavation of a grave, pay for the coffin, pay for the headstone, pay for the interment, etc.
Maybe add that you can keep the ashes (have fun shopping for a decorative urn?), or bury the ashes under a favorite tree in your yard, if you think they'd be comforted by having their pet's remains physically close to them.
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