In the Lord of the Rings why did Sauron blow up when he only had his fingers cut off?
In the book, there is no indication that Sauron exploded. In fact, it is not clear at all within the Lord of the Rings itself that Sauron died when the Ring was removed.
From Letter 131:
But to achieve this he had been obliged to let a great part of his own inherent power (a frequent and very significant motive in myth and fairy-story) pass into the One Ring. While he wore it, his power on earth was actually enhanced. But even if he did not wear it, that power existed and was in 'rapport' with himself: he was not 'diminished'. Unless some other seized it and became possessed of it. If that happened, the new possessor could (if sufficiently strong and heroic by nature) challenge Sauron, become master of all that he had learned or done since the making of the One Ring, and so overthrow him and usurp his place. This was the essential weakness he had introduced into his situation in his effort (largely unsuccessful) to enslave the Elves, and in his desire to establish a control over the minds and wills of his servants. There was another weakness: if the One Ring was actually unmade, annihilated, then its power would be dissolved, Sauron's own being would be diminished to vanishing point, and he would be reduced to a shadow, a mere memory of malicious will.
So without the Ring, Sauron is not in any way diminished. His power still exists; he isn't weakened by the absence of the Ring. The Ring is a tool for Sauron to focus his power through to accomplish things he wouldn't otherwise be able to do - not because his strength was insufficient to the task, but because he didn't have the "tool" to do so. On the other hand, assuming material form is a natural ability of the Maiar.
From The Council of Elrond:
I beheld the last combat on the slopes of Orodruin, where Gil-galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword, and took it for his own.
Here, though it isn't absolutely clear, it seems Isildur cut the Ring off Sauron's hand after Sauron was overthrown.
Again, in The Shadow of the Past:
It was Gil-galad, Elven-king, and Elendil of Westernesse who overthrew Sauron, though they themselves perished in the deed; and Isildur Elendil's son cut the Ring from Sauron's hand and took it for his own.
Appendix A also repeats the mantra that "Sauron was overthrown, and the One Ring was taken from him."
And again, in the Rings of Power and the Third Age:
But at last the siege was so strait that Sauron himself came forth; and he wrestled with Gil-galad and Elendil, and they both were slain, and the sword of Elendil broke under him as he fell. But Sauron also was thrown down, and with the hilt-shard of Narsil Isildur cut the Ruling Ring from the hand of Sauron and took it for his own. Then Sauron was for that time vanquished, and he forsook his body, and his spirit fled far away[...]
You might then ask what "overthrown" means. Certainly it means Sauron was defeated, but was he dead? The text of Of the Rings of Power seems to imply that Sauron was not utterly vanquished until Isildur cut off the Ring, and Isildur claims shortly after that he dealt the Enemy his death-blow.
On the other hand, Tolkien frequently uses the same phrases to mean someone's death. For example:
I threw down my enemy, and he fell from the high place and broke the mountain-side where he smote it in his ruin.
Another example off the top of my head, Morgoth is said to have been overthrown by the Valar in the Akallabeth; in that case, Morgoth was beheaded and his spirit ejected from Arda.
So it seems that at most, Isildur finished Sauron off, and whatever wound he inflicted in taking off the Ring killed Sauron. At the least, Isildur cut the Ring off Sauron's corpse. Possibly some of the confusion comes from the fact that Maiar are immortal. Sauron's body could be totally broken and ruined, and he would still not be "dead." Perhaps Sauron did not choose to abandon his body until the Ring was taken from him, even though his wounds up to that point were already "fatal."
Either way, he didn't blow up. That was creative license in the movie.
The visual "explosion" of Sauron in the film adaption of The Lord of the Rings was creative license. There are no notes about exactly how he was vanquished. But the representation is certainly not implausible. At this point in the timeline of Middle-Earth, he was unable to take a fair form. Years before the Battle of the Last Alliance, Sauron was vanquished in the Fall of Numenor. Though he survived, he was unable to take any fair form ever again. By the time of the Last Alliance, his spirit had regained the power to produce a form. But it was the power of the Ring that was fueling his spirit and his form. Once this was separated from him, he was diminished and his spirit fled, taking many years to once again gain power, though the form he took is debated. As noted in answers to this question,
...in a tale which allows the incarnation of great spirits in a physical and destructible form their power must be far greater when actually physically present. ... Sauron should be thought of as very terrible. The form that he took was that of a man of more than human stature, but not gigantic. ~ The Letters of J. R. R. Tolkien
But that does not relate to his form at the time of the Last Alliance. Then, he "blew up" because he was separated from the Ring, wherein was contained much of his power.
Short answer -it looked good on screen :) It's a pity that movie did not show up the last fight between Sauron and GilGalad and Elendil. In book there is stated both died when defeated Sauron. GilGalad died probably because Sauron touched him by his burning hand.
I think the battle at the beggining was more a retelling, so the explosion was likely a Metaphor for Sauron being destroyed. What always bothered me is how the ring supposedly fit over his plate-armour but also fitted a Hobbit's finger. Something's not quite right there.