If you are at a loss to entertain young children during a rainy day in summer, you can always play them a DVD of Frozen – the animated movie which is proving to be the biggest hit that Walt Disney has ever produced. They may have seen it before, but that probably doesn't matter. Sam Goldwyn once said of film audiences: "If they liked it once, they'll love it twice."

In the case of Frozen, multiply that by a score. My grand-daughters, aged five and four, have seen the movie 40 times since it came out on DVD in March, but they are always ready for a 41st viewing, singing along with the music, and in flawless American children's voices.

It is a huge, worldwide hit, especially adored in the UK, Japan, Malta, Ireland, Denmark and Venezuala. It's taken a billion and a half dollars at the box office, garnered many awards, and won high praise from the critics.

So what is the secret of its success? When I watched it first in the cinema, I thought it rather dark. In the early part of the story, the two little princesses, Elsa and Anna, are tragically orphaned when their parents are drowned at sea. I found this upsetting, but young children seem to be able to accept such sad details quite philosophically.

The story is about the relationship between the two sisters. But Elsa, the elder, has "cryokinetic" magic powers which enable her to create snow and ice. In childhood, this gift is used playfully, but after causing a serious snow accident, Elsa becomes introverted and withdrawn, and shuts out her younger sister from her companionship. Elsa's cryokinetic power is to be admired, but also feared. "There is beauty in it, but also great danger," says one of the trolls, who are magic healers. For Elsa, the power has brought her guilt and isolation.

As in all classic fairy tales and legends, a journey must be undertaken, obstacles overcome, and sacrifices made before truth and redemption bring a satisfactory resolution.

Little sister Anna – hurt by her sister's seeming rejection – accepts a marriage proposal from the first man she encounters, Prince Hans. She sings an aria: 'For the first time in forever ... I'll be noticed by someone.' Yes, this is the adolescent's need for reassurance and attention. Elsa, meanwhile, though crowned queen of her country (a Nordic state called Arendelle) has shut herself up in her ice palace, "where I can be where I am without hurting anybody". She's "alone but free".

She sings about coming to terms with her true self, and throwing off the shackles of the expectations of others. This motif has been interpreted as a gay 'coming-out' affirmation by some LGBT cultural analysts.

Idina Menzel, who is the voice of Princess Elsa, thinks the message of the movie is a feminist one. "It's essentially about sisterhood," she has said.

Sisters, either biological or social, may be competitive, but they are also protective of each other. The theme of sisters who love one another, but are separated by a misunderstanding, is a touching one, and I'm near enough to tears each time I watch Frozen (only three times, as yet), notably in the sequence when Anna sacrifices her life for her sister. (Never fear – she will be recalled to life by love.)

The Disney production team has said the success of Frozen is a surprise, and yet, it's obvious why it has such wide appeal: the music is infectious, and the animation, a mixture of hand-drawn and computer-generated images, is marvellous.

Jennifer Lee, who conceived the story – partly inspired by Hans Anderson's The Snow Queen but substantially changed – has brought to life real, beguiling characters.

Anna, the fiery redhead, is so spirited and persistent: she won't take no for an answer. "You don't have to protect me!" she cries. "I'm not afraid!"

Kristoff, the good guy who helps Anna, is drawn with eyes rounded and sympathetic, whereas the prince who turns out to be a villain, has narrowed, mean eyes. The nuanced expressions on the characters' faces is something akin to great acting. And it's visually stunning – the research the animators did in Norway, even observing the effect of light on snow and ice, has paid off.

Like all stories that enchant children, Frozen is both innovative and conservative. The trolls hand out conventional wisdom: "People make bad choices if they're mad or scared or stressed".

Self-sacrifice is a strong theme: the snowman, Olaf, who yearns for something he cannot have – warmth, and the heat of summer – is ready to melt himself down for Anna's sake. "Love is putting someone else's needs before your own."

Power is dangerous – Elsa's terrible power sets off an eternal winter in her kingdom, but people can learn to use power judiciously, and power may also be put to the service of others.

Risks must be taken for the sake of love and understanding, but everything must be done with correct procedure: Elsa must be crowned queen in church by the ecclesiastical authority.

Children learn about psychology from fables and legends. Einstein once said: "if you want your children to be intelligent, read them fairytales."

The psychoanalyst Bruno Bettelheim believed such stories endure and catch on because they contain metaphorical truths and universal wisdoms.

I am not entirely sure everything in Frozen is metaphorically true: can an act of true love really thaw a frozen heart?

Yet we want the messages in this story to be truthful and that is at the heart of its appeal.

Belfast Telegraph