Species are exactly as real, and as unreal, as rivers, for many of the same reasons. It is true that every river (species) is different, and it is hard to describe their boundaries, and their beginning and ending points are often unclear. Also, there are some arbitrary aspects about rivers, like when we call them a river or a stream, or how we divide and name segments of a river.
When two rivers merge into one (or one species splits into two), it is very hard to determine exactly when that happens. Yet, it does happen, and it is pretty easy to see after the fact (but not in the close details of water molecules or individual genes at the point of merger). But the concept of a river is very useful, and I have floated down many rivers that seemed pretty real to me. I also regularly encounter and interact with a lot of species that seem quite real to me. All of these difficult attributes I mentioned are true of all historical lineages: species, countries, corporations, rivers, families, or almost anything to which we assign a proper name (or could).
Yet, why are biologists perplexed about species? Clearly, life is not distributed continuously, but into lineages of historically related individuals that share a common reproductive (or sometimes, ecological) boundary. I just can't see how one can make any sense out of life without understanding and accepting the reality of species. It would be like trying to understand the economy while denying the reality of corporations, or global politics while denying the reality of countries.