Adding recessed lights to existing ceiling fan connection
At the Light

Connect the white wire that leads to the recessed lights, and the white wire from the fan/light to the white wire in the ceiling box.
Connect the the black from the recessed lights, to the black ungrounded (hot) wire that's controlled by the dimmer.
Connect the red wire from the fan/light, to the red ungrounded (hot) wire that is not controlled by the dimmer.
Connect all the grounding conductors.

At the switch

Connect the LINE wire (probably black, but depends on manufacturer) from the dimmer, in with the black wire from the feeder cable.
Connect the black wire from the armored cable, to the switched (probably red) wire from the dimmer.
Connect the ground wire from the dimmer, with the other grounds in the box.

This should allow the dimmer to control the recessed lights, and the remote to control the fan/light.
Of course this assumes that power comes to the switch first, and one of the wires in the cable is always hot.
Based on the info you've provided, it should look something like this.


According to the documentation for the dimmer, the black lead is LINE and the red lead is LOAD.