Meaning of assumptions regarding continuum of agents in economics models
Economic models often have a finite number of agents. In such models, agents are commonly named using the positive integers (i.e. 'natural numbers'): so the set of agents is $\{1, 2, ..., n\}$.

Other economic models have an infinite number of agents. For example, the number of agents might be assumed to be the same as the number of positive integers. In that case, the set of agents is $\mathbb{N}=\{1, 2, ...\}$. It is said to have a 'cardinality' (or size) of aleph-null.

Finally, some economic models have even more agents. In the examples you mention, each agent is named using a real number in $[0, 1]$; so the set of agents is the set of real numbers in $[0, 1]$. This set is said to have cardinality aleph-one and can be shown to be 'larger' than the set of natural numbers (in the sense of 'larger' defined by Cantor).

Of course, this is not very realistic -- the justification, presumably, is to make the mathematics easier. In particular, it allows for integration (e.g. to find total consumption in the economy) as opposed to summation.