The story of Rumpelstiltskin is quite well known, but I can't recall what his motivation for wanting the human child. The question can be extended to Goblins in general. (Although in the case of the Tengu, who sometimes would eat the babies, hunger for delicacies seems a sufficient motivation;) But for faeries in general who don't eat the infants, what is the impetus for baby stealing?
'Something that lives is dearer to me than all the treasures in the world!'
So says the little man when offered treasures by the Queen in return for not taking her child as she'd previously agreed.
But when she laments and weeps because she can't bear to be separated from her baby he pities her, and gives her the riddle of his unknown name to find.
Because he's a "little man", a manikin, he can't find love and marriage. It seems that he has no need of wealth because of his knack of creating gold out of straw but he can't make a life with his magic. So he's desperately lonely and craves companionship, a baby to care for, someone to be grateful to him and so take care of him in return, like children and parents' love for eachother. He clearly has sympathy for the Queen, feeling with her the unhappiness at losing her precious child, and offering her a reprieve, so it seems the child is precious to him as well.
Perhaps this story was told to prepare a girl for the end of childhood: Her father is trying to marry her off and the prospect of leaving her family home and beloved parents and going to a strange house to do God-knows-what with a strange adult man is stressful for the girl. A child often has a precious mannequin or doll that is imbued with character, power and intent by virtue of child-like imagination, and this entity, in the imagination of the child, would inherit the child's love of parents and feeling of being loved by parents through being loved like a baby by the child. This doll has magical creative power that is able to cope with any looming unnameable fear that lies beyond a child's natural range of comprehension (in the absence of sex education), and child and doll enjoy a secret relationship, involving secret language/sounds/names. But Mistress Miller's creative power, having a real child of her own, comes into play when she is able to relinquish her childhood (and her doll) symbolized by uttering the name of her half-imagined childhood friend. No longer needed after the birth of her own child the mannequin uses it's magical power to self-destruct and he conveniently tears himself in two!
It's a theory.
In most tales, as in Rumpelstiltskin, there is no reason given for fairies or goblins wanting to take human children. I would like to note, though, that in some tales similar to Rumpelstiltskin, the goblin actually wants the girl herself. In the English "Duffy and the Devil" the Devil wants to carry a girl off (presumably to Hell) and in the German "Doubleturk" the goblin wants to marry the girl. (Other Rumpelstiltskin variants are here.)
In a lot of changeling myths, it's suggested that fairies look for beautiful children, perhaps because they want good breeding stock. (See "The changeling of Spornitz.") In these tales they may treat the children as their own and even transform them into beings like them. The Brothers Grimm mentioned nixies who were originally stolen human children ("Nixie Changelings from the Saal River.")
Very frequently, fairies or goblins want human servants, as in "The Daughter of Peter de Cabinam" (from the 13th century) and "The Fairy Dwelling on Selena Moor."
The idea of fairies stealing away humans was very much a danger in European folklore. Fairies found humans to be desirable, whether for labor, or beauty, or another reason. I believe it was a way to explain sudden deaths and illnesses. It was not something to be explained so much as something to be avoided - by protecting pregnant women and newborns, by not angering the fairies, not intruding on their domain (like mushroom circles) and warding them off from your own home by hanging horseshoes over doors or other practices.
If I may put in a shameless plug, here is a list where I tried to collect as many examples as I could: "Why do Fairies Steal People Away?"
A very interesting question. This is my stab at it.
The text itself does not answer this question. The “Encyclopädie des Märchens” states on pg. 1250 in a section on promising/selling the child to the devil that “only seldom are the motivations of the demon, what he wants aims to do with the promised child, for wanting the child, further explained. The Devil naturally has a human soul in mind in the Christian sense… other supernatural beings are in search of a partner (man or woman) for their own children….not seldom are the handed over children employed in service of the demon.” The EM goes on to say that the origin of this lies in human sacrifice as shown in the Bible and mythology." My guess, and it is only a guess, is that Rumple S. wants the child as a servant.
In KHM #39 II “Regarding The Wichtelmännern” (AKA “The Elves” in Hunt’s 1884 translation), they get a young girl so that she can take a child out of the baptismal. The implication is that it is their child that she takes out of the baptismal.
Mythology changed over the centuries, especially under christian influence. A large crack can be observed around 900-1000 CE when large parts of Europe were christianized, i.e. Sweden (Harald Bluetooth), Norway (Harald I. Fairhair), Vladimir I. (Kievian Rus), ....
Up from this time mythological beings that personify the psychological archetype of Anima, the most fundamental female archetype, like fairies and elves received evil charactert traits in mytholgoy, that they did not possess before. For example the elf shot also appeared in mythology not before the 10th century CE.
The birth of changelings (disability) and the stealing of babies (possibly a reference to a miscarriage) is a very female topic and so with the beginning of the (male dominated) christian era the 'guilt' for these events was blamed on beings (fairies, elves,...) that personify the most fundamental female archetype, the Anima. Very likely this was not the case in the pre-christian era.
So basically the impetus for fairies to steal children (trigger a miscarriage) or for elves to hurt humans by elf-shots is a late christian insertion into mythology (blaming the guilt for these events on fairies and elves)
In the native pre-christian Indo-European mythology the guilt for events that were harmful to society was not blamed predominantely on women because a fundamental concept of the native pre-christian Indo-European mythology is that the universe is in a constant balance and order (Rta (Vedic), Veritas (Roman), Urda (Norse), ...)
With christianization the Indo-European priestesses and shamen turned into witches and what happened to witches is well known.