Let us assume for the moment that the quality of a theory should be judged based on the accuracy of its predictions, and not, for example, on how well it explains the world. We can certainly envision doing science without explanations. In such a world we would simply observe a regularity in some phenomenon and make the prediction that events would continue in basically the same fashion (of course we would have to conduct some statistical analysis on the data as well, in order to make the theory accurate). And if observations confirm that prediction then it would be a good theory, and just as good as a theory that made the same predictions but also explained why the regularity existed. What role then do explanations play?
Well, when theorizing about a single phenomenon, explanations may not really be necessary. When we study a single case there is really no basis for choosing one explanation over another. Imagine studying only the decay of one single isotope into another. We can easily conceive of several possible explanations. One explanation might invoke forces within the nucleus, one might appeal to tiny gnomes, and one might describe the decay as simply a fundamental property of the atoms. Without investigating other phenomena there is no real way to distinguish between them. And since atomic decay occurs in a fairly regular way all of these theories could be equally accurate in making predictions, so why should we care?
But science cannot usefully progress by studying isolated phenomena. There are simply too many starting points from which events may develop to study each one in isolation. What we need then are theories that can make accurate predictions in a wide range of cases. If we are presented with such a theory we can test it in a number of different cases (although not in all cases), and if it proves reliable it is reasonable to act under the working assumption that it is reliable in all cases, until proven otherwise. Such theories are the goal of science, not predications about isolated phenomena.
A vital step in this process is developing candidate theories that make predictions for a wide range of situations. Of the possible explanations of atomic decay I mentioned above only the appeal to forces within the nucleus has the possibility to be extended to other cases. Appealing to gnomes, or simply assuming that atomic decay is a fundamental property, doesn’t give us a basis for predicting the decay of other isotopes. These other explanations can’t be extended to the more general case simply because, unlike the case of the single isotope, we simply can’t make enough observations to develop possible laws of decay directly from them. Appealing to gnomes only worked because we could come up with exact predictions based on our observations alone, an appeal to gnomes without such data simply has no basis for making predictions. When considering isotopes in general there are simply too many cases to come up with laws from the observations alone. Even though we should make as many observations as we can in order to construct better candidate theories we simply can’t look at all the possibilities.
This is where the need for an explanation arises. We can construct possible explanations based on the limited observations we have made, and on the basis of our proposed explanation make predictions for the cases that we have not observed. Now we could come up with general laws without explanations, but without observations covering every possibility we would be essentially shooting in the dark, as far as the unobserved cases were concerned. So even if explanations aren’t strictly necessary for a theory to be successful (in the sense of accurate) they are needed to arrive at the broad theories that are actually useful.
You can read a related post here.