The phrase kangaroo care denotes a method of caring for a baby, especially a premature one, which emphasises the importance of holding the infant in skin-to-skin contact with a parent for as long as possible each day.

The earliest occurrence of the phrase kangaroo care that I have found is from When Baby’s First Home is the Neonatal Intensive Care Unit (NICU), by Sharon McNeil, RN [= Registered Nurse], CNNP [= Certified Neonatal Nurse Practitioner], in a full-page advertisement for the Berkshire Medical Center, published in The Berkshire Eagle (Pittsfield, Massachusetts) of Sunday 1^st April 1990:

Parents are strongly encouraged to visit often and to hold their baby as soon as his/her condition warrants, regardless of what the baby is “connected” to. Mothers who want to breastfeed are very much supported in doing so as this has untold benefits for both baby and mother, especially if the baby is small and sick. Parents begin assisting with and learning their baby’s care from very early on. Our Special Care Nursery at BMC has begun encouraging mothers (and even fathers!) to do “kangaroo” care with their infants. This is a method of caring for babies by keeping them bundled next to their mother’s (or father’s) body in skin-to-skin contact for extended periods of time. This simple and natural way of nurturing has been shown to do much to relieve many of the physiologic signs of stress in these babies and to provide much needed warmth.

In Kangaroo’s Pouch Inspires Care for Premature Babies—about the Neonatal Intensive Care Unit at the Brigham and Women’s Hospital in Boston, Massachusetts—, published in The New York Times (New York City, New York) of Wednesday 10^th June 1992, Elisabeth Rosenthal explained:
– the etymology of the phrase kangaroo care;
– the origin of the method denoted by this phrase.
Her text also contains:
– the phrase kangaroo visit, denoting a visit during which a parent holds their infant in skin-to-skin contact;
– the verb kangaroo, meaning: (of a baby) to receive kangaroo care; (of a parent) to give kangaroo care:

Inspired by marsupials, who nurture their young in a pouch, “kangaroo care” is challenging the high technology philosophy of intensive care for babies.
[…]
Kangaroo care allows for the bonding between parent and child that is usually precluded in intensive care units. And from those bonds may stem physical benefits: some studies have suggested that premature babies who “kangaroo” have fewer breathing problems, gain weight faster and maintain a more stable body temperature than babies confined to incubators, or isolettes.
[…]
Nurses who have supervised kangaroo visits observe that babies become much calmer with their parents and waste fewer calories in the kind of restless fidgeting that occurs in incubators.
[…]
Three weeks ago, Edna Scannell and Earl McAllister began kangaroo visits with their daughter Lucretia, who was born April 22, at 1 pound 12 ounces. Since the care began, she has started to gain weight, and although no one can prove that the skin-to-skin visits have contributed to her progress, her parents are converts.
“At first we couldn’t even touch her, and that was just awful,” Ms. Scannell said. “Once I started kangarooing, the improvement was amazing.”
Kangaroo care was developed in Colombia about a decade ago, when doctors were facing a serious shortage of incubators for premature babies, who are unable to regulate their own body temperature.
Taking a cue from kangaroos, the doctors devised slings that allowed mothers to carry their infants constantly next to their chests.
The program was so successful that the United Nations Children’s Fund promoted it throughout the underdeveloped world. And recently a few nurses at leading hospitals in the United States have sought to adapt it to their working environment.
[…]
Before kangaroo visits, parents could sometimes hold the baby, but wrapped in blankets and under warming lights. “The babies would squint and the parents felt hot,” Ms. Field [the nurse in charge of the Newborn Intensive Care Unit at the Brigham] said. “It was hardly a natural visit.” Such visits were always short, since the babies started to fidget, the blankets disrupted their monitors and their body temperature proved impossible to maintain.
In contrast, parents’ chests have proved to be impeccable natural thermostats, said Jennifer Wallace, a clinical nurse specialist at the Brigham. In fact, the nurses have observed that if a baby’s temperature drops, the parent’s actually rises to respond to the baby’s need.