Example: You fire a 4AD of photons, a single line-item "weapon system" repesenting a bank of four photon tubes, at a target in the forward arc and range 5". You roll 1d6 for each of them, rolling all four dice at once, to "hit" the target. As the target is less than one-half max range 1/2 of 15" is 7.5"), the to-hit roll required in 4,5,6. Any "Natural 6" will by-pass shields and do internal damage. (Note: If the weapon is "Accurate", add +1 or +2 to the die roll, but only a "Natural 6" by-passes shields).
The dice come up 2,4,5,6. The "2" is a miss, so we can ignore that. The "6" by-passes the shields and goes internal, but we'll hold off on that for now. The "4" and "5" hit the shields. The Multihit-4 means you just did 8 points of sheild damage. As the target's shields were at 5 due to previous damage, the target marks off the last five boxes and the extra 3 points goes internal. Now you add in 4 points for the photon that rolled a "6", for a total of 7 points internal.
You then roll 7d6 on the Attack Table on page 7 to see what the damage was. Every "1" does nothing, but the rest do one point of damage to the damage track. Every "6" does a point of damage -and- rolls a d6 on the Critial Hits Systems Table on page 8. (Note: If the weapon is "Precise", add +1 to the die roll, so any "1" does damage, and any roll of "5" or "6" is a critical hit.)
You rolled 1,1,2,3,3,5,6 and thus did five points of damage plus one critial hit. Next, roll one d6 on the Systems Table. You rolled a "5" which is a Crew hit. Because phtons are "Devistating +1", the this becomes a Level 2 critical. The target loses one Marine for the Level 1 critial, and two more for the Level 2 critical. Additionally, the ship takes another point of damage on the damage track. That critical hit is marked "Escalate", so at the end of the game-turn, the taget must attempt to stop the cascade damage.
I can tell you what I do in my games, but it perhaps should not be considered official until it appears in the FAQ.