Here's a way you could model the problem. Suppose you have clients numbered $i=1,2,\ldots,M$, and each client has an associated probability $p_i$ which is the probability that they don't ruin a batch of messages with spam if they are included in the batch.
The simplest thing to do would be to say $p_i=0$ for spammers, i.e. they always spoil the batch. Then you could say the likely spammers were clients who never participated in a good batch of messages.
More elaborate, you could say that each spammer has the same probability of passing $p_i=\sigma$ and each non-spammer passes with probability $p_i=\nu$.
Then for a given set of clients $C$ in a batch, the probability of a good batch of messages is
$$
G_C = \prod_{i\in C} p_i
$$
and the probability of a bad batch of messages is $1-G_C$.
Given a set of batches with client sets $C_1,C_2,\ldots$, the likelihood of getting the realized good/bad labelling back is
$$
P = \prod_{C_j~\mathrm{good}} G_{C_j} \prod_{C_j~\mathrm{bad}} (1-G_{C_j})
$$
We can look for the maximum likelihood condition by maximizing $\log P$
$$
\begin{align}
L &= \log P = \sum_{C_j~\mathrm{good}} \log G_{C_j} + \sum_{C_j~\mathrm{bad}} \log (1-G_{C_j}) \\
& \simeq \sum_{C_j~\mathrm{good}}\sum_{i\in C_j} \log p_i - \sum_{C_j~\mathrm{bad}}\prod_{i\in C_j}p_i
\end{align}
$$
to a first order approximation assuming $G_C$ is close to zero for bad batches. Then
$$
\frac{\partial L}{\partial p_i} \simeq \sum_{\mathrm{good}}\frac{1}{p_i}-\sum_{\mathrm{bad}}\frac{G_{c_j}}{p_i}
$$
and we can look the best spammer/non-spammer classification by starting with and arbitrary labelling (say all non-spammer, e.g.), then reclassifying the clients with largest negative derivative as spammers, and/or those with largest positive derivative as non-spammers, then repeat as necessary until you converge to a (local) maximum.
Some variations of this approach would be to consider the probabilities on a message-by-message basis to account for different volumes, or to use something like the EM algorithm to also try to determine $\sigma$ and $\nu$. But since you asked for layman's terms I think I should stop here.
