So, first off, he didn't invent a new alphabet: Glagolitic derived from (for the most part) a heavily modified Greek cursive script, and the later Cyrillic alphabet (with which Cyril is identified but didn't create) derived from (again, mostly) a less-heavily modified Greek uncial script.

For the most part 11th century Cyrillic uncial looks a lot like 4th-7th century Byzantine Greek uncial, and differences mostly hinge on the differences of Byzantine Greek to Slavic in terms of their sound systems. Greek β had the value of /v/ by Byzantine times, and thus Cyrillic В is also /v/, and a modified form Б was created for /b/ (this is clearer when you look at the uncial forms here). Early Slavic had a distinction between /z/, /dz/, and /ʒ/ (as in pleasure), necessitating the creation of ѕ /dz/ and ж /ʒ/ to go along with zeta. Early South Slavic also had /ts/ Ц, /tʃ/ Ч, /ʃ/ Ш, and /ʃt/ Щ, which all necessitated new characters, whose origin is obscure but most likely Hebrew.

The remaining characters introduced into Cyrillic represented vowels (at the time of their creation), and include Ъ, Ꙑ, Ь, Ѣ, Ѧ, Ѫ. Ъ and Ь originally represented vowels something like those in English book and bit, but shorter, and Ꙑ represented a vowel like bit but with the tongue set a bit further back in the mouth. Ѣ represented a vowel like English cat. Ѧ and Ѫ represented nasal vowels, the first like in French vin and the second like in French bon.