It came from the fans of White Wolf's World of Darkness games. "Splat" is another name for the asterisk character ('*'), which is often used as a placeholder or "wild card" in a name by technical types of people. Someone somewhere starting referring to all of WW's various Clanbook/Tribebook/Guildbook/Kithbook supplements for their various games as "*books", pronounced "splatbooks."
From there, the term expanded out into the fans of other publishers' game lines that also followed a publishing scheme that offered player-facing supplements based on in-setting organizations or classes of characters. TSR had already been publishing The Complete X's Handbook supplements, and it naturally was applied to them in quick order. Between the vast market share held by the combination of White Wolf and TSR in the 90s, the term was virtually guaranteed to become commonplace—and it did.
Shannon Appelcline looks at the origin of the term "splatbook" in Designers & Dragons: The 90s in the chapter on White Wolf:
Splatbooks had been around since almost the dawn of roleplaying. […] However, no one had previous put out splatbooks as consistently and in such volume as White Wolf did. Clanbook: Brujah (1992) was the first. […] White Wolf would go on to produce splatbooks for all of their initial lines, and the term "splatbook" was eventually coined for White Wolf's releases.
…and again in the appendix entry, "10 Things You Might Not Know About Roleplaying in the ’90s: 1. The Splatbook Cometh":
The biggest change for the industry in the ’80s may have been the appearance of the splatbook. […] However, no one followed in GDW's footsteps until the very end of the ’80s. Only then did TSR kick off a new era of RPG publications with their PHBR series of Complete Handbooks for various AD&D classes (1988–1995). White Wolf followed and typically gets more credit for the splatbook revolution because they published a lot more. Starting with Clanbook: Brujah (1992) for Vampire: The Masquerade, each World of Darkness got its own series[…] In fact, White Wolf's extensive sets of splatbooks generated the term, with *book (pronounced "splatbook") referring to a book with a title noun at the start like “Clan” or “Tribe”.