Why is EXPTIME in EXPSPACE? What is the intuition behind EXPTIME being inside EXPSPACE?

When space complexity is usually smaller than time complexity or in the worse case, they are equal
As Yuval Filmus states in the comments, any TM (with one tape and one read-write head) whose time complexity is bounded by a function $t$ also has its space complexity bounded by $t$ since, in $t$ steps, the TM can only scan as most as $t$ many tape positions. In terms of complexity classes:
$$\textsf{TIME}(t) \subseteq \textsf{SPACE}(t)$$
$\textsf{P} \subseteq \textsf{PSPACE}$ and $\textsf{EXPTIME} \subseteq \textsf{EXPSPACE}$ is immediate from this. One of the greatest concerns of complexity theory (most of which are yet unsolved questions) is determining under which conditions this is a proper inclusion or not.