In Lear's poem "The Duck and the Kangaroo," the Duck admires the Kangaroo's ability to hop all around. Desiring to leave the pond and see the world, the Duck asks the Kangaroo for a ride on his back. The Kangaroo has "but one objection" to the Duck's request, and that is the discomfort the Duck's cold, wet feet would give. The Kangaroo suggests that carrying the Duck's chill-producing feet would probably give the Kangaroo "the roo-Matiz."
"Rheumatiz" is "rheumatism" spoken in a country or backwoods dialect. Rheumatism was a catch-all term for joint and muscle pain, especially the type that worsens in cold, rainy weather. The word comes from the root "rheum," meaning fluid, because before the advent of modern medical knowledge, such symptoms were thought to come from excess bodily fluids ("rheum") seeping into the joints.
Lear, of course, uses a clever misspelling of the word—"roo" instead of "rheu"—to suggest a kangaroo version of the malady. He also breaks the word after the first syllable to create a rhymed couplet with the word "Kangaroo" in the next line.
Thus "roo-Matiz" is a version of cold-and-damp-induced muscle and joint pain that kangaroos in Lear's fantasy world can suffer from. The Duck, sensitive to the Kangaroo's concerns, dons worsted wool socks before the two embark on their three round-the-world trips, so the Kangaroo remains a healthy and happy traveler.