A normal bootstrap ACM takes hot high pressure bleed air, takes out some of the heat while still at high pressure, compresses it some more to boost the pressure and temperature again, takes out more of the heat while the pressure is still elevated, then lets it expand to drop the pressure, and thus the temperature, to a much colder temperature than where it started. The bootstrap effect(from the saying about pulling yourself up by your bootstraps), comes from amplification effect you can achieve by exploiting the kinetic energy in the flow with the compressor and turbine to amplify the temperature changes.
A reverse bootstrap works the process backwards, as a way to get an amplified (bootstrap) cooling effect from ambient ram air instead of high temperature/high pressure bleed. These systems are used for cooling secondary components via heat exchangers by enhancing the effect of ram air, instead of providing large volumes of cold air for cooling a cabin directly.
The air is basically passed through the ACM backwards, from the ram air inlet directly to the turbine, leaving the turbine expanded to a lower temperature than it was at the ram air inlet. This chilled air leaving the turbine goes through a heat exchanger that cools whatever it was you wanted cooled off (like an electronics rack). The air picks up heat from the heat exchanger, cooling the electronics you wanted cooled, more efficiently than if it was just ram air alone.
The job of the compressor in this backwards flow cycle is only to create suction in the ducting upstream of it, that is, the ducting from the turbine discharge, to the heat exchanger, to the compressor inlet. The suction enhances the temperature drop you get from the ram air expanding and dropping its pressure through the turbine. The turbine itself extracts energy from the from the ram flow in the work it does to drive the compressor, further helping with the chilling effect.
So in the reverse bootstrap cycle, instead of amplifying hot high pressure bleed flow's pressure and temperature to get a bigger temperature drop than otherwise, and discharging expanded cold air for directly for air conditioning, instead you amplify the cooling effect of ram air that you pass directly through the expansion turbine first, and the compressor's job is just to generate suction, further dropping the pressure of the turbine discharge to cool it more. And the compressor's discharge just gets dumped overboard having done its job.
To summarize, the reverse bootstrap effect is the use of the kinetic energy in the ram air flow to drive the turbine to drive the compressor, which creates suction to amplify the temperature drop at the turbine.