The $t$ in $t$-design is essentially a measure of how good a job the set of gates does in terms of randomising a state (the larger t, the more random, with properly random requiring the infinite limit). Often, you want to compute the average of some function over all possible pure input states, which is equivalent to fixing the input state and averaging over all possible unitaries. However, averaging over all possible unitaries is a pain, and is unnecessary if the function you want to compute is simple enough. If the function you want is a polynomial of degree t or less in terms of the coefficients of the input state, it is sufficient to average over a set of gates that comprise a t-design.
Another way of thinking about this is, instead of a degree t polynomial, you can talk about calculating a linear function of t copies of the input state. This is more like you would do in an actual experiment.
As for what makes the Clifford group a 2-design, I guess you just have to sit down and do the maths. There's a proof in section A.1 of this paper. For the special case of the Clifford group on a single qubit, let S be the set of 1-qubit Clifford gates. Then you need to show that
$$
\sum_{s\in S}s\otimes s|00\rangle\langle 00|s\otimes s\propto\mathbb{I}+\text{SWAP}
$$
The critical thing here is that there’s 2 copies of the state that we’re averaging over.
