According to the Internet:
There are two main forms of the prefix "in-" in English.
in- (1)
prefix meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant), from Latin in- "not," cognate with Greek an-, Old English un-, from PIE *ne "not" (see un- (1)).
in- (2)
element meaning "into, in, on, upon" (also im-, il-, ir- by assimilation of -n- with following consonant), from Latin in- "in" (see in). In Old French this often became en-, which usually was respelled in English to conform with Latin, but not always, which accounts for pairs like enquire/inquire. There was a native form, which in West Saxon usually appeared as on- (cf. Old English onliehtan "to enlighten"), and some verbs survived into Middle English (cf. inwrite "to inscribe"), but all now seem to be extinct. Not related to in- (1) "not," which also was a common prefix in Latin: to the Romans impressus could mean "pressed" or "unpressed."
The use of "in-" in "increase" corresponds with the second definition.
increase (v.)
early 14c., "become greater in size or number; to cause to grow, enlarge," from Anglo-French encress-, Old French encreiss-, present participle stem of encreistre, from Latin increscere "to increase, to grow upon, grow over, swell, grow into," from in- "in" (see in- (2)) + crescere "to grow" (see crescent). Latin spelling restored 15c. Related: Increased; increasing.
In case you wanted to see the entry about the word "in", to which the second definition is related, here it is:
in
Old English in (prep.) "in, into, upon, on, at, among; about, during;" inne (adv.) "within, inside," from Proto-Germanic *in (cf. Old Frisian, Dutch, German, Gothic in, Old Norse i), from PIE *en "in" (cf. Greek en, Latin in "in, into," Old Irish in, Welsh yn-, Old Church Slavonic on-). As an adjective from 1590s.
The forms merged in Middle English. Modern sense distinction between in and on is from later Middle English. Sense of "holding power" (the in party) first recorded c.1600; that of "exclusive" (the in-crowd, an in-joke) is from 1907 (in-group); that of "stylish, fashionable" (the in thing) is from 1960. The noun sense of "influence, access" (have an in with) first recorded 1929 in American English. In-and-out "copulation" is attested from 1610s.