On many (American) phone keypads, 1 is blank, 2 is A, or B, or C, 3 is DEF, 4 is GHI, 5 is JKL, 6 is MNO, 7 is PQRS, 8 is TUV, and 9 is WXYZ.
So for FLOWERS you'd see that the F is on 3, L is on 5, O is on 6, etc. to get 356-9377. Note that both R and S are on 7, they map to the same number, so different words are likely to all map to the same number, like "Roses" and "Ropes".
Companies sometimes use longer or shorter numbers: you might see a number written with left-padded numbers, like "1-800-1FLOWER", for which you'd dial 1-800-135-6937. You might also see longer words, like 1-800-ARRANGEMENTS..you could type in 1-800-277-2643 63687, but the extra letters would just go into the recipient's PBX (like dialing an extension) and get discarded.
A lot of phones come with the English alphabet on the 12 keys, three letters to a key. Similar to older phones that had a physical dialpad. You could then dial the "word" you want by looking at the number on which the correct letter is. That's one way these systems could work.
The use for vanity numbers is a later development. Originally the three digit exchange "number" in North America was identified by a three letter code. On a phone with no dial you'd tell the operator the exchange codeword and the four digit number within the exchange if you didn't want to manually pulse the switchhook. For automatic switch dialed calls, the dial needed to also be marked for these letter codes so you didn't need to use a separate reference for the corresponding digit.
It certainly added wealth to the phone number holder/assigner.