What do I need to do to raise my "Speed" rating?

As noted by Arcane, "speed" in this context means the total number of turns it takes to complete all of the battles. As noted on the wiki, the way to get an A is to beat all of the battles in 28 turns or less:

Speed Rank  Multiplier  Total Turns
    A          1.5x      Turns ≤ 28
    B          1.4x        29-31
    C          1.3x        32-35
    D          1.2x        36-39
    E          1.1x      Turns ≥ 40

This number does not appear to be affected by the number of maps, so it's much harder to get an A rank on the harder difficulties. With only 3 maps, that's an average of ~9 turns/map, while with 7 maps you only get 4 turns/map.

A suggestion that may help in this category is to use units with more mobility and range. Using cavalry units will help you get into position easier, and bow/dagger/magic users will be able to hit enemy units from a farther distance. A team of armored units are unlikely to clear these battles efficiently enough to get an A for speed.

Note that each team you lose loses you 5% bonus from the survival category, which stacks multiplicatively. An A for survival with an E rank for speed is 1.2*1.1 = 1.32 which is slightly worse than a D survival and C speed (1.05*1.3 = 1.365), so it'll be worth losing a few weaker teams for speed as long as you can at least get a C.

You need to focus on completing your maps in the fewest turns possible to increase your speed. On average I complete most maps in 2-3 turns, 4-5 max. This gives me a consistent A rank for speed. Since the multipliers stack, don't worry to much about surviving and focus on quick wins. If your team dies and you have to use a second team as long as you kept up the speed and had a bonus ally you'll suffer only a small drop in your score