Why did the Europeans feel they were on a “civilizing” mission in the time from 1750 to 1900? Did race play a role?
Unquestionably, race was a major factor in the European "civilizing" missions of the time period mentioned. Europeans had for centuries considered themselves superior to other peoples because they were Caucasian, an idea that was embraced in the concept of Social Darwinism, that the Caucasian race was naturally superior to another:
There are underlying, and largely irresistible, forces acting in societies which are like the natural forces that operate in animal and plant communities. One can therefore formulate social laws similar to natural ones. These social forces are of such a kind as to produce evolutionary progress through the natural conflicts between social groups. The best-adapted and most successful social groups survive these conflicts, raising the evolutionary level of society generally (the 'survival of the fittest.)
The fact that Europeans were Christians re-enforced this feeling of racial superiority; in fact Europeans did not distinguish between "civilizing" and "Christianizing" other peoples, as they considered them to be one and the same. As early as the seventeenth century, the coat of arms of the Massachusetts Bay Company had depicted an American Indian with the slogan "Come over and help us." This was actually a biblical reference indicated that the Indians wanted Christianity, that is Western Civilization. There were other superficial reasons for "civilizing" the world, including commercial profit; however there would have been no effort to force European society on others were it not for the underlying belief that white European society was the only true "civilization."
Christianity and racism were so closely connected that they were inseparable. To suggest that racism was not a major factor in European imperialism is to accept an overly simplistic approach.
It is hard to know for sure if race played a role, but it seems likely that it was one of the reasons Europeans (and Americans) felt this way.
Europeans felt that they needed to "civilize" the rest of the world largely because the rest of the world was technologically backwards compared to them. This led to Europeans feeling that they were superior to the other people in many ways. They felt that their religion was superior and that their social and governmental systems were superior. They also felt that they were superior racially, but it is hard to be sure if racism came first or if it simply came about because of the other kinds of feelings of superiority. In other words, did Europeans feel that they were superior because of race or did the feel superior because of technology and then come to think that race was connected to that? We can't know the answer for sure.
Overall, then, Europeans felt superior in terms of economics, society, government, and religion. Because they felt superior, they felt they could and should civilize the other people of the world.