The high-level view of how it works:

https://www.wartsila.com/marine/products/exhaust-treatment/c...

The Wärtsilä carbon capture solution works in five main stages: pre-conditioning, absorption, desorption (or stripping), liquefaction and storage.

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The final stage is storage. The liquid CO2 is pumped through pipes to onboard tank(s), where it is stored until it can be offloaded.

Carbon capture on vessels while they sail has been achieved behore by Mitsubishi in 2021. https://www.mhi.com/news/211020.html The challenge is where to store that CO2 while sailing as each ton of fossil fuel burnt creates 3 tons of CO2. The average ocean going vessel burns approx 200 tons of fossil fuel per day = 600 tons of CO2. On a 10 day journey that results in 6000 tons of CO2. That requires a lot of space and additional weight.