Vrddhi is the higher of the two strengthening grades the Indian grammarians describe, and for the front series it produces this vowel. Where guna raises i to e, vrddhi raises it past e to ai, and the grade is called for by a specific set of suffixes, most notably the ones that form patronymics and adjectives of belonging. A very large part of the Sanskrit vocabulary of relation and derivation runs through that operation: the name of a descendant, the adjective naming what pertains to a place or a school, and a wide class of abstract nouns all show the vrddhi vowel where the underlying stem shows the simple one.
The result is that this letter is met most often at the head of a word rather than inside it, because the strengthening applies to the first syllable of the stem. That in turn means the independent letter, rather than the attached mark, is the form a reader sees. In the Prakrit of the Ashokan edicts the picture is different: the compound vowels are less stable in Middle Indo-Aryan, and forms with a plain e where the older language had ai are ordinary, so the letter is written far less often in the earliest inscriptions than a Sanskrit text would lead a reader to expect.
The two grades are also what makes a Sanskrit dictionary usable. Entries are listed under the root in its plain form, and a reader who meets a derived word has to strip the suffix and lower the vowel to find it. That operation is taught before anything else in the language, and this letter is one of the two vowels a beginner learns to reduce on sight. A word opening with it is very often a derivative of a root opening with the simple front vowel, and recognising the relation is the difference between reading a text and decoding it.