First, a shorthand definition of Evolution is heritable change and not 'progress' (which is a rather loaded word). Also, evolution is not 'supposed' to do anything, and can lead to both an increase or a decrease in diversity, since (local) diversity is simply the net result of speciation rate and extinction rate (if we exclude colonization of new species from other areas). Therefore, I think the question is framed in the wrong way.
Second, there are many reasons why biologists might want to halt species extinctions, as hinted in the other comments and answers. Most obviously, conservation biologists want to prevent man-made extinctions which are due to how we have influenced/modified/destroyed the natural environments. I don't think conservation biologists in general are determined to halt all 'natural' extinction. However, most of the time it is really hard to pinpoint what would be a 'natural' extinction. Either way, the current rates of extinction, which have been estimated to lie 100-1000 times (IUCN, 2007) above background rates (some say it's closer to 10000 times higher), indicate that most extinctions are man-made.
The rationales people use for preserving biodiversity can however differ. Three distinct categories are:
1) For human survival and/or economy, e.g. through ecosystem services or resources we rely on.
2) For human welfare, e.g. recreation or esthetical reasons.
3) Ethical considerations, e.g. inherent value of species or to preserve the unique evolutionary history of the system.
Another way to structure these is to label 1) & 2) as instrumental values and 3) as inherent values. The instrumental values can then be subcategorized as services, goods/resources, information (e.g. genetic to be used in medical research), and esthetical/spiritual values.
Since the direct value of individual species are often (usually) uncertain (irrespectivly of the direct rationales used) the precautionary principle is often invoked to justify species conservation. Also, numerous examples exist of how aspects of biodiversity are related to human values (some recent examples are Garibaldi et al. 2013 and Gamfeldt et al. 2013), such as production or services, which can also be used to justify both conserving biodiversity or individual species. However, other aspects of biodiversity are functional diversity, which focus on preserving functional groups of species, or phylogenetic diversity, focusing on taxonomic coverage and uniqueness (evolutionary history), instead of 'overall' biodiversity (see e.g. Flynn et al, 2011 for a discussion of these concepts). Under these perspectives less focus is placed on individual species.