I have no idea who started using the abbreviation “Xmas” or the reason for it, but I suspect it was simply to abbreviate the term by using the English letter ‘X,’ which represents the Greek letter chi, the first letter of Christos (Cristo) for Christ. If you will note, the first Greek letter of christos, the chi, looks very much like our English letter X. Christos means “anointed one” and comes from the verb chrio (criw), “I anoint.”
The Microsoft Bookshelf dictionary has this to say:
Xmas has been used for hundreds of years in religious writing, where the X is understood to represent a Greek chi, the first letter of CrstoV, “Christ”; in this use it is parallel to other forms like Xtian, “Christian.” But the letter X, or especially x, is nowadays more frequently interpreted as a mathematical variable than as a Greek letter, as indicated by the common pronunciation of the form Xmas as ( ). Thus, while the word is etymologically innocent of the charge that it omits Christ from Christmas, it is now generally understood only as an informal shortening. In an earlier survey 88 percent of the Usage Panel rejected the use of Xmas in writing (The American Heritage® Dictionary of the English Language, Third Edition copyright © 1992 by Houghton Mifflin Company. Electronic version licensed from INSO Corporation. All rights reserved).