To clarify the spatial–temporal evolution patterns and climate-driven mechanisms of carbon sinks of forest ecosystems under climate change, we calculated the net ecosystem productivity (NEP) of forests in the Guangxi region using remote sensing and meteorological data from 2000 to 2023. By employing trend analysis, spatial clustering, the Hurst index, and climate contribution evaluation, we analyzed the spatial and temporal changes, sustainability, and the relative contribution of climate impacts on forest carbon sinks. The results are as follows: The carbon sink capacity of forests in Guangxi increased continuously from 2000 to 2023, at a rate of 3.57 g C·m⁻²·a⁻¹, reaching 39.19% higher in 2023 than in 2000. The carbon sink capacity was higher in the southwest and lower in the northeast, with hotspots mainly located in evergreen/deciduous broad-leaved forest areas. The Hurst index indicates that 84.44% of regions are likely to maintain this increasing trend, suggesting stability in forest carbon sink function. The climate contribution rate to forest carbon sinks was moderate, with significant temporal fluctuations. Temperature governed annual variation in forest carbon sinks, influencing up to 36.37% of the area. The annual average contribution rate of climate change to forest carbon sinks was 30.28%, but there were temporal fluctuations and spatial heterogeneity. Over time, climate contributions had a positive driving impact; however, extreme climate events tended to produce a negative effect. The pattern of forest carbon sinks in Guangxi showed a “heat sink-coupling” phenomenon, with 16.23% of the hotspots of forest carbon sinks coinciding with temperature control zones, highlighting the enhancing effect of temperature rise on carbon sinks against a background of water and heat synergy. This study provides a scientific basis for the assessment of forest carbon sink potential and climate suitability management in Guangxi.