c. 1300, "a chessboard, checkerboard," from Anglo-French escheker "a chessboard," from Old French eschequier, from Medieval Latin scaccarium "chess board" (see check (n.1); also see checker (n.2)). The governmental sense of "department of the royal household concerned with the receipt, custody, and disbursement of revenue and with judicial determination of certain causes affecting crown revenues" began under the Norman kings of England and refers to a cloth divided in squares that covered a table on which accounts of revenue were reckoned by using counters, and which reminded people of a chess board. Respelled with an -x- based on the mistaken belief that it originally was a Latin ex- word.
also formerly expence, late 14c., "action of spending or giving away, a laying out or expending," also "funds provided for expenses, expense money; damage or loss from any cause," from Anglo-French expense, Old French espense "money provided for expenses," from Late Latin expensa "disbursement, outlay, expense," noun use of neuter plural past participle of Latin expendere "weigh out money, pay down," from ex "out, out of" (see ex-) + pendere "to hang, cause to hang; weigh; pay" (from PIE root *(s)pen- "to draw, stretch, spin"). For the financial sense of the Latin verb, see pound (n.1).
Latin spensa also yielded Medieval Latin spe(n)sa, the sense of which specialized to "outlay for provisions," then "provisions, food" before it was borrowed into Old High German as spisa and became the root of German Speise "food," now mostly meaning prepared food, and speisen "to eat." Expense account is from 1872.