Old English berstan (intransitive) "break suddenly, shatter as a result of pressure from within" (class III strong verb; past tense bærst, past participle borsten), from a West Germanic metathesis of Proto-Germanic *brest- (source also of Old Saxon brestan, Old Frisian bersta, Middle Dutch berstan, Low German barsten, Dutch barsten, Old High German brestan, German bersten "to burst").
The forms reverted to brest- in Middle English from influence of Old Norse brestan/brast/brosten, from the same Germanic root, but it was re-metathesized late 16c. and emerged in the modern form, though brast was common as past tense through 17c. and survives in dialect.
In Old English "Chiefly said of things possessing considerable capacity for resistance and breaking with loud noise; often of cords, etc., snapping under tension; also of spears, swords, etc., shivered in battle" [OED]; in late Old English also "break violently open as an effect of internal forces." Figuratively, in reference to being over-full of excitement, anticipation, emotion, etc., from c. 1200. Transitive sense ("to cause to break, cause to explode") is from late 13c. Meaning "to issue suddenly and abundantly" is from c. 1300 (literal), mid-13c. (figurative). Meaning "break (into) sudden activity or expression" is from late 14c. Related: Bursting.
1610s, "act of bursting, a violent rending; a sudden issuing forth," from burst (v.). Meaning "a spurt, an outburst" (of activity, etc.) is from 1862. Jane Austen, Coleridge, Browning use it in a sense of "a sudden opening to sight or view." The earlier noun berst (early Middle English) meant "damage, injury, harm."