I know you have explicitly asked for an intuitive explanation and to leave out the formal definition, but I think they are rather related, so let me recall the definition of typical set:
$X_1, X_2 ,... $ are i.i.d. random variables $\sim $ $p(x)$ then the typical set $A_\epsilon^{(n)} $ with respect to $p(x)$ is the set of sequences $(x_1,x_2,...,x_n) \in \chi^n$ with the property
$$2^{-n(H(X)+\epsilon)}\le p(x_1,x_2,...,x_n) \le 2^{-n(H(X)-\epsilon)} \tag{1}$$
This means that for a fixed $\epsilon$, the typical set is composed of all the sequences whose probabilities are close to $2^{-nH(X)}$. So in order for a sequence to belong to the typical set, it just has to have a probability close to $2^{-nH(X)}$, it usually does not though. To understand why, let me rewrite the equation 1 by applying $log_2$ on it.
$$H(X)-\epsilon\le \frac{1}{n}\log_2\left(\frac{1}{p(x_1,x_2,...,x_n)}\right) \le H(X)+\epsilon \tag{2}$$
Now the typical set definition is more directly related to the concept of entropy, or stated another way, the average information of the random variable. The middle term can be thought as the sample entropy of the sequence, thus the typical set is made by all the sequences that are giving us an amount of information close to the average information of the random variable $X$. The most probable sequence usually gives us less information than the average. Remember that, the lower the probability of an outcome is, the higher the information it gives us will be. To understand why let me give an example:
Let's suppose that you live in a city whose weather is highly likely to be sunny and warm, between 24°C and 26°C. You may watch the weather report every morning but you wouldn't care much about it, I mean, it is always sunny and warm. But what if someday the weather man/woman tells you that today will be rainy and cold, that is a game changer. You will have to use some different clothes and take an umbrella and do other things that you usually don't, so the weather man has given you a real important information.
To sum up, the intuitive definition of the typical set is that it consists of sequences that give us an amount of information close to the expected one of the source (random variable).