This page describes the history of bengali language, not the script. For that, see here. Some examples of old bengali can be found here.
Bengali is a language of the Indo-Aryan branch of the Indo-European family of languages. Its Indo-European roots can be traced to the initial split in middle Indo Aryan languages into a NW dialect which grew into Gandhari, the central dialect which grew into Sauraseni, an eastern dialect which became Magadhi and a Southern dialect which lead to, for example, Maharashtri. Bengali probably arose out of the Magadhi or Ardhamagadhi prakrts, through mAgadhI apabhraMza. Its earliest examples are the caryagItis. (A slightly different view claims that the origins of Punjabi, Hindi, and Rajasthani actually share a distinct origin than the rest of the indo-aryan languages.)
However, Bengal was not always indo-european. (See the description of aryanization of bengal for related information.) This group of languages are first attested in India in the Rgveda, the language of the earliest parts of which probably dates back to the copper-bronze age punjab and west, 1400–1200 BC. This language already shows effect of a substratum/adstratum which gave rise to most of its agricultural terms, names of wild plants and animals, and such phenomena as the retroflex consonants. The later iron age atharva-veda, the language of which goes back to 1200–800 BC, still talks about the east as on a distant horizon, whereas the vAjasanIYa saMhitA talks about the language of magadha as being harsh, and aitareYa brAhmaNa talks about the easterners as dasyu. Even the later (800–500 BC) brAhmaNas like zatapatha brAhmaNa calls them AsUrya. It is only in the kauSitakI brAhmaNa that we find easterners (probably still not bengal, though) going west to learn the proper language, though baudhAYana dharma sUtra still call them azpRzya. It is only a few centuries before the christian era, during the time of rAmAYaNa and mahAbhArata, that the east seems to have come in the indo-european fold. Later, pANini mentions the eastern languages are spoken with slightly different grammar; and pataJjali observes that the r sounds often change to l in this speech (a character possibly from austroasiatic languages), already a feature of mAgadhI prAkRta, attested for the first time in brAhmI script during the maurya period (3rd-2nd cent BC) in mahAsthAnagaDh.a.
The standard development of the language breaks the periods as prAcya form of late old Indoaryan (700 BC), Early middle Indoaryan (300 BC), transitional middle Indoaryan (1 CE), second middle Indoaryan (300 CE), mAgadhI apabraMsha or late middle Indoaryan (800 CE), old bAGlA (1100 CE), early middle bAGlA (1400 CE), late middle bAGlA (1600 CE), and new bAGlA (1800 CE). In this chronology, distinctly eastern pronounciations, such as its default half-open back vowel, insertion of the i sound, the merging of the sibilants, the l/r alteration, and the desibilization of kS, and some features of nominal declinations like the -r genitive, -e locative, and some verbal features like -l- in the past and -b- in the future are already noticeable in the early middle Indoaryan period.
Before the indo-aryan languages, we find a Austroasiatic (mon-khmer or muNDA) substratum giving words like gaNDa, gu~D.i, khA~khA~, khA~khA~r, bA~khAri, bAdur, kAni, jAG, TheG, Tho~T, pAgol, bAsi, chA~c, chA~ctala, chocJA, kali, choTa, peT, khos, jhAD., jhop, cimil, karAt, dA~, bAigan, pagAr, gaD., baraj, lAu, lebu, kalA, kAmrAGga, dumur, puNDra, tAmlitti, kabdAk > kapotakSa, dAmdAk > dAmodar, Dhe~ki, moTA. It also gives the dA of shiyaldah, jhinaidah, bA~shadaha, and possibly the phenomenon of doubling seen in pulinda-kulinda, mekala-utkala, accha-vaccha, uNDra-puNDra-muNDa, kosala-tosala, anga-vanga, kalinga-tilinga, takkola-kakkola. These people were probably rice farmers and had bananas, eggplant, lemon, betel, coconut, jAmburA, kAmarAGgA, Dumur, turmeric, betelnut, pomegranate, and cotton. They probably knew of sheep (used for blankets), bow and arrows, elephant, birds, crow, crabs, sickle, saw etc. They probably knew of metal working, metal and stone arms, farm equipment and utensils, about games like guTi and pAsha, about weaving. In addition to rice and vegetables, food probably consisted of yava, wheat, fish, sheep, pigs, chicken. The animals they knew included oxen, cows, buffalos, sheep, elephants, camel, pigs, goats, chicken, dogs, peacocks, and probably horses. (See here for domestication of plants and animals.)
A later Dravidian source of words like the -jola, -jota, -jotika, -hitti, -hitthi, -vitthi, -hist, -histhi, -gadda, -gaddi, -pola, -vola, -handa, -vada, -kunda, -kundi, -cavati, -cavada, -D.A, -guD.i, -juli, -juD.a in place names and words like bhiTA, kuNDu, khokA-khuki, is also seen, especially in the west. The language of the north and east has borrowed from the Tibeto-Burmese group of languages.
In addition, Bengali absorbed from the Farsi, Turkic and Arabic languages under the muslim rulers and English under the British.
Up to history of Bengal