Why drop the “i” in “explanation”?

Solution 1:

Expanding on ΜετάEd’s answer, the alternation between long /e:/ and short /a/ (or reduced version thereof, schwa or a mid-high vowel) is now frequent in Modern English (and harks back to the Great Vowel Shift).

So, alongside expl[e:]n ~ expl[ə]nation, you also have:

• ex'pl[e:]n ~ expl[æ]natory
• ins[e:]ne ~ ins[æ]nity (insane ~ insanity)
• in[e:]ne ~ in[æ]nity (inane ~ inanity)
• n[e:]tion ~ n[æ]tionality (nation ~ nationality)
• [e:]ble ~ [ə]bility (able ~ ability)

As you hunched and ΜετάEd pointed out, there is an orthographic oddity to explain, in that its /e:/ sound alternates with a lax vowel, but it is, orthographically, ai. The nearest I can come to like examples is:

• retain ~ retention (this too is an etymological oddity, examples like contend ~ contention, intend ~ intention_, and so on, would lead one to expect retend ~ retention)
• inveigh ~ invective (where the origin of eigh is again etymologically odd, coming from invehere)

As the parenthetic comments make clear, these are as much oddities as the case you identified.

Solution 2:

The question should be: where did that i come from.

If we look at etymonline we find the following: (emphasis mine)

explain (v.)
early 15c., from Latin explanare "to make level, smooth out;" also "to explain, make clear"

Originally explane, spelling altered by influence of plain. Also see plane (v.2). In 17c., occasionally used more literally, of the unfolding of material things: Evelyn has buds that "explain into leaves" ["Sylva, or, A discourse of forest-trees, and the propagation of timber in His Majesties dominions," 1664]. Related: Explained; explaining; explains.

So it seems that the form explane, probably pronounced much like the current version, lost its final e and to reflect the pronunciation in the spelling, an i was added.