Your current circuit is 240V split phase, which means both wires are hot and there is no neutral. Therefore, to switch it on and off, both hot wires need to be disconnected together. Your switch is only capable of switching one of them.
Simplified image of your current wiring, with colors re-marked for consistency:
I can think of two possible solutions:
Option 1: Separate Switch and Dimmer
Use another dimmer, in addition to your current switch. Your switch is really two devices in one -- a switch to cut power to the lights, and a signaling device to tell the lights how much to dim. However, there's no reason these two devices need to be combined. You can keep using your current two-pole switch for the on/off function, and get a dimmer that only has the 0-10V dimming function (Example) -- it will only have the purple and grey wires. You don't need to put it next to the current switch, though you will probably want to for convenience's sake.
Option 2: Re-wire Circuit for 120V
Since your new lights use much less power than your old ones did, you don't need the extra power that a 240V circuit provides anymore. One of the current hots can be replaced with a neutral wire, and then the combination switch/dimmer you bought will be compatible. However, since NEC doesn't allow re-labeling wires at this size, you would need to pull out one of the current hot wires and replace it with white or grey wire. At the breaker panel, the remaining hot will land on one side of the breaker (or replace it with a single-pole breaker), and the new wire will land on the neutral bus. Then, at the switch location, simply wire-nut the neutral wires together.
Diagram of wiring re-configured for 120V:
Separately from all of this, you'll need to run dimming control wires to each fixture. Check the labeling on the fixture to see whether you can run the wires in the same conduit as the power (Class 1 wiring) or if you need to use a separate wiring method (Class 2).