Fundamental differences in the radio properties of red quasars (QSOs), as compared to blue QSOs, have been recently discovered, positioning them as a potential key population in the evolution of galaxies and black holes across cosmic time. To elucidate their nature, we exploited a rich compilation of photometry and spectroscopic data to model their spectral energy distributions (SEDs) from the UV to the FIR and characterise their emission-line properties. Following a systematic comparison approach, we infer the AGN accretion, obscuration, and host galaxy properties in a sample of ~1800 QSOs at 0.2<z<2.5, classified into red and control QSOs and matched in redshift and luminosity. We find no differences in the average SEDs of red and control QSOs, other than the reddening of the accretion disk expected by the selection. Moreover, no clear link can be recognised between the QSO reddening and the interstellar medium or the star formation properties of their host galaxies. We find that the torus properties in red and control QSOs are strikingly similar, suggesting that the reddening is not related to the torus and orientation effects. Interestingly, we detect a significant excess of infrared emission at rest-frame 2-5 um, which shows a direct correlation with optical reddening. To explain its origin, we investigated the presence of outflow signatures in the QSO spectra, discovering a higher incidence of broad [OIII] wings and high CIV velocity shifts (>1000 km/s) in red QSOs. We find that red QSOs that exhibit evidence for high-velocity winds present a stronger signature of the infrared excess, suggesting a causal connection between reddening and the presence of hot dust in QSO winds. We propose that dusty winds at nuclear scales are potentially the physical ingredient responsible for the colours in red QSOs, as well as a key parameter for the regulation of accretion material in the nucleus.
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