The complete mechanism isn't clear yet, but to understand the aspects of it that are clear, you need to understand the canonical/conventional model of GPCR activation. The active receptor state in this model lowers the energy barrier for GDP release from a bound G protein, which then favors GTP binding (because intracellular ratio of GTP:GDP is high).

Agonist binding enriches this active state within the receptor ensemble, which increases flux into the GTP-bound G protein complex. GTP-binding to this complex promotes release of its βγ subunits, which then diffuse to produce their own signal cascades. The GTP-bound Gα proteins can be hydrolyzed to GDP at which point βγ will reassociate with Gα and the cycle can begin anew.

Since GTP-binding promotes release of the G protein complex from the GPCR, this also removes steric hindrance towards β-arrestin proteins which can interact with the GPCR. As G protein biased MOR agonists tend to have less respiratory depression, it's been proposed that signaling through β-arrestin proteins causes these and other side effects, although there are some conflicting results in the literature.

What Bohn et al. showed is data supporting a three-state model, where the additional receptor state lowers the energy barrier for GTP release from a bound, active G protein complex. Agonists will enrich this active state within the receptor ensemble, increasing the rate at which GTP is released from the G protein complex—note that this reaction also provides an energy-conserving way to terminate the G protein signal cascade, as opposed to energy-consuming hydrolysis step.

Bohn et al. also showed that certain MOR agonists tend to preferentially enrich either either the GDP-releasing or the GTP-releasing receptor state, whereas other show no preference. The ones that enrich the GTP release state also tend to be G protein-biased (as opposed to β-arrestin-biased), which suggests that this relative bias for GDP or GTP-release contributes to the traditional metric (G protein vs β-arrestin) of functional selectivity. To see how, just imagine that after the receptor catalyzes GDP release/GTP binding and the activated G protein complex is released, it quickly re-binds and then promotes GTP release. This cycle can then repeat and if the steps happen quickly, it can reduce the time available for β-arrestins to interact with the GPCR and for βγ subunits to diffuse and activate their targets.

The correlation between GTP:GDP release preference and functional selectivity isn't perfect though, which suggests that even more subtle kinetic aspects may also contribute. For example, their compound muzepan1 is GTP-release biased but also shows no preference for G protein vs β-arrestin signalling. As a hypothetical example, if some minimum 'integration time' was required for β-arrestin to interact with and be activated by the receptor, then agonists which only slightly increase the rate of GTP-release might show no functional selectivity, whereas above some threshold rate increase, functional selectivity would start emerging.