"Dial M for Murder" meaning

It's a take-off on "dial O for operator".

For example, in Tennessee Williams' Streetcar Named Desire (1947), we have

Blanche: How do I get Western Union? — Operator! Western Union!
Stella: That's a dial phone, honey.
Blanche: I can't dial. I'm too —
Stella: Just dial O.
Blanche: O?
Stella: Yes. "O" for Operator.

Since dialing "O" gets you the operator, one might expect by analogy that dialing "M" would get you the murder (somehow ... the details are not filled in).

It's a reference to the letters you sometimes see next to the numbers of a telephone pad (a rotary dial in those days). When telephone numbers were first introduced people were much more comfortable with letters than digits and would commonly give mnemonics for at least the first few digits of the number. In New York all the numbers in Flushing Meadows started with 35 corresponding to "FL". Glenn Miller had a song called "Pennsylvania 6-5000" (736-5000, apparently the number is still in use). It would not be uncommon when the movie (and earlier the play) came out for people to give part or all of a phone number using letters.