A Solow model problem I am trying to solve asks me to assume that the economy consists of identical agents of mass 1. I have never heard of this term before, what does this mean?
In this example, "mass" means the same as "length". Let me try to explain it with a simple example.
If you work in a continuous setting, you have an infinite number of agents, which lie on a line between $0$ and $M$. Let's say individual consumption is given by the function $C(i)$, for all agents $i \in [0,M]$.
For total consumption, you have to integrate: $C = \int_0^M C(i)di$ (total consumption is the area under the graph). $M$ is the mass of all agents. In your example, $M$ is equal to 1.
This is not done for realism. Continuous agents are usually used when you want to introduce heterogeneity in a model. For instance, in the standard New Keynesian model (see, e.g., Walsh, Monetary Theory and Policy, chapter 8), households don't face one price, but a distribution of different prices, and hence also different demand.