If you keep a constant power setting then the way to decrease your speed is to pitch up. Simply, this will increase your rate of climb as more of your total vector is ‘upward’ rather than ‘forward’. The CDU is assuming a constant power setting during the climb so will naturally calculate a higher rate of climb for a lower airspeed.
Initially, pitching up will trade airspeed for altitude. Going from a climb at 300 knots to a climb at 250 knots means pulling back & increasing the rate of climb while the speed decreases.
For a close-in fix to the climb profile, this is often sufficient to rapidly gain several hundred feet above the FMC prediction and solve the situation.
For a steady ...
I think you have it backwards, it is not that a slower airspeed produces a steeper climb, it it that a steeper climb, given the same power, will cause a slower airspeed.
If the only constraint you're setting the FMS is the airspeed to maintain for a constant thrust, then it will pitch up more, increasing the rate of climb, which will slow the airspeed.