In the context of experimental evolution, replicate lines are simply separate experimental (or control) lines the are established from the founder population at the beginning of the experiment. In this paper, the authors must have established 10 separate experimental lines that then evolved in different carbon dioxide levels. For example, 10 separate aliquots of algae are drawn from the same source population and then separately subjected to the agent of selection. These replicate lines should remain separate and not be mixed or crossed.
Replicate lines are useful in experimental evolution for a few reasons. Because you are studying a population that is experiencing a selection pressure, you expect that genetic changes are in response to that pressure. However, changes can also occur due to drift. Replicate lines (and a consistent response to selection among them) is important to ensure that changes are a response to selection rather than to some other force (drift, mutation). Some drift is bound to happen when the lines are founded, because you are selecting a subset of the population, which is actually a good thing. Each line gets a random subset of genes which then evolve together. This potentially allows discovery of different genetic pathways to the same adaptive response.
The background and lots of examples are in Garland and Rose, Experimental Evolution (UC Press, 2009).