This is, of course, a delicate subject; a Catholic believer's answer to your question would be different to that given by a secular historian, and would acknowledge the perceived holiness of the child witnesses and the nature of the miracles that are attributed to them. My answer comes more from the secular and comparative perspective, and that needs to be borne in mind.

First, for those who may not be too familiar with the Fatima visions and the various subsidiary phenomena attached to them, here's a quick overview of the case.

The Fatima apparitions took place in 1917, and were reported by three young children, who were then aged 10, 9 and 6, and who were working as shepherds at a remote spot called Cova da Iria in rural central Portugal, north of Lisbon. The main witness (and it is very typical of Marian visions in general that there is a main witness, with all that may imply from a psychological perspective) was Lucia de Jesus dos Santos, aged 10 when the series of events began. She was the only one of the three children to claim that she interacted with the Virgin. It is at best highly doubtful whether the two subsidiary witnesses, Francisco Marto (who was 9 in 1917) and his sister Jacinta actually saw and heard any, much less all, of the things that Lucia described.

The first significant event alleged to have taken place at Fatima dates to 13 May 1917, and it took place in daylight while the children were tending a flock of sheep in a pasture two miles to the west of the town of Fatima itself. The generally accepted version of events is that the three children were frightened by a flash of lighting and fled down the hillside, only to encounter the dazzling apparition of a beautiful woman, radiating white light, whom they saw standing amid the branches of an oak tree (a slightly surreal touch which, again, is not untypical of Marian visions). Lucia approached, and was told (in the local Portuguese dialect the children spoke) that the lady would return five more times, on the 13th of each month; at the end of that period, she would reveal who she was, and there would be a miracle.

This sequence of events gives us the first clue to the potency and significance of the Fatima apparitions. They were a sequence of events, predicted in advance, which allowed ever-increasing crowds (50 in June, more and more as the months passed, until there were tens of thousands present) to gather at the spot in the hope of witnessing a miracle. The vision itself, however, was not vouchsafed to the members of the crowd. Lucia and the two other children positioned themselves by the same tree, and Lucia explained to the crowd that the vision had reappeared, gliding down from the sky into the branches; she would recount the details of the conversation she had with the vision. Those who believed her account had their faith rewarded, and also, importantly, each successive "appearance" further cemented the children's reputation as divinely appointed intermediaries, and Lucia's standing, in effect, as a seer – the successful predictor of miraculous encounters – as well.

By the end of the six-month period, in October 1917, the crowd that had assembled was estimated to number 70,000. It was on this occasion that a significant number of witnesses among the crowd claimed to have witnessed the celebrated "dance of the sun," in which the sun was reported to have left its spot in the sky, and (depending on who gave the account) whizzed around like a Catharine wheel, or fallen towards the assembled crowd. The exact number of witnesses of this event is very uncertain; all we can really say is that this vision was certainly witnessed by fewer than half of those present. Nonetheless, that means that several dozens or hundreds or thousands of witnesses – most of them religiously inclined, and in a state of great expectation, it is true, but some of them sceptics – believed that they had seen something at Fatima. From this perspective, the Fatima events were undoubtedly the most heavily witnessed and reported series of Marian visions there have ever been, and if the upper range of witness numbers is accepted, there's a good case to be made that the final vision was the most widely-witnessed miraculous event that has ever taken place. This, too weighs heavily in explaining their importance to the Catholic church.

Now, it's also important to recognise that much of what has been written about Fatima dates to much later than 1917. The famous "Three Secrets of Fatima," for instance, were not written down until more than 20 years later. Indeed, Lucia gave at least four increasingly elaborate versions of events, in four separate memoirs. For example, she later claimed that the 1917 visions were not the first she had had. There were, she said, another six in 1915 and 1916, involving the apparitions of an angel and a ‘statue made of snow.’

Sceptics draw attention to the problems raised by these competing accounts. There are, for example, substantial discrepancies between the accounts that Lucia gave immediately after her experiences in 1917 and the version of events she set down in her memoirs in 1942, two decades after she had taken her vows as a nun. It was in this year that, as well as doubling the total number of visions she had previously claimed, Lucia set down for the first time two of the three famous ‘secrets of Fatima’ said to have been entrusted to her and her two companions. These included a horrifying vision of hell and prophecies concerning the future of Russia.

Anyone who is not a believer in the reality of the visions is likely to suspect that much or all of this material must be of questionable reliability. As the author Kevin McClure, one of the most sympathetic and sensitive of authorities on such matters, observes,

Spending over 20 years in the quiet and seclusion of a religious house, cut off from the normal run of daily life, undoubtedly reliving the events of 1917 time and again, is not the best background for recalling precise details of what must have been deeply traumatic events in the life of a young and innocent child. It would have taken a staggering degree of objectivity for those memoirs to have been an accurate record of what actually happened.

The more sceptical Joe Nickell harbours graver doubts about Lucia’s reliability as a witness, labelling her ‘petted’ and ‘spoiled’, not to mention the self-proclaimed possessor of ‘a gift for telling stories’, and pointing out that her own mother is said to have labelled her ‘nothing but a fake who is leading half the world astray’.

In answer to your question, though, I'd point to the following factors – in addition to those already mentioned above – as crucial in raising the Fatima visions in the estimation of the Catholic public.

Further, how much did Portugal's Estado Novo propaganda machine contribute to the Church's recognition and how true are the remarks that Fatima's acceptance by the Church and the Portuguese authorities are mostly derived from the Church and Portugal's stance towards communism and the fact that Fatima's message is clearly and literally anti-communist and critical of the USSR?