The prefix be- has a wide range of meanings and applications. It can be added, forming transitive verbs, to nouns (befriend), adjectives (belittle), and other verbs (bespeak) and it can help turn nouns into participial adjectives (witch → bewitched; suit → besuited).
Prefixing a word with be- often lends the sense ‘about, around, all over’ or ‘completely’. It can also intensify it, as in the line ‘Snails, much despised, bekicked, and becrushed’ in George Kearley’s natural history book Links in the Chain (1863). Or it can suggest affecting or afflicting something greatly, as in bestench (1568) ‘to afflict with stench’.
The prefix was common in Old English, appearing in words like befealdan ‘fold round’ and behātan ‘promise’ (examples are from Burchfield’s The English Language) and becoming part of prepositions like before, behind, below, beneath, and beyond. In Middle English be- continued to spread, being added also to imports from French and other Romance languages: becalm, beguile, belabour, besiege.