If both the A and B input is held HIGH, neither diode will have a voltage drop, there will be no current flow; no current flow means no IR drop across the resistor, hence V will appear at the output (a HIGH).
If either or both the A and/or B input is grounded (a logical LOW), either or both diodes will be forward-biased and current will flow between V and ground through one or both diodes. This current will cause an IR drop across the resistor equal to about (V-0.7v) leaving the remainder (0.7v) at the output for a logical LOW. It's not the best circuit in the world, for that's a sloppy LOW, assuming silicon diodes. Using germanium diodes, the output would be closer to 0.3v for a logical LOW, maybe 0.1 to 0.2v using hot carrier (Schottky) diodes. So, it does work as an AND gate: if both A AND B is HIGH, the output is HIGH; in all other cases, the output is LOW. This also translates to a DeMorgan equivalent AND: If either A or B goes LOW, the output goes LOW; in all other cases the output is HIGH.
Dean