The Russians view aircraft carriers as air defence equipment.... think of it as a PVO CVN.
It provides air borne eyes (AWACS) and air reach (fighter aircraft).
It wont necessarily be used to invade countries... the two large helicopter carriers they are currently building will allow them to deploy land forces where needed, but also to assist in humanitarian situations, or to support a friendly country suffering from western abuse (ie like Syria or perhaps Libya if they could have gotten there sooner...).
They are not going to travel the world sinking enemy ships and stealing resources like oil or gas, but the west has a significant naval presence that could interfere with Russian trade with her customers.
The west has already seized Iranian ships and gifted the weapons on board to Ukraine... how long before they seize oil tankers and steal the contents for their own purposes.
You can't stop them with submarines... submarines become a liability during peace time because you can either sink the enemy ships or threaten to sink the enemy ships... not a great range of choices.
Compare that with having an aircraft carrier nearby and having a flight of 4 Su-33s armed with R-77 and R-37 air to air missiles and perhaps a couple of Kh-31s and Kh-35s just to say move along to those US or UK ships trying to stop your naval commerce.
And in 10 years time imagine 4 Su-57s with the Kh-31s that are mach 3.5 240km range anti ship missiles supplanted with similarly sized Kh missiles with a rocket scramjet propulsion that fly at perhaps mach 9 and 400km maybe...
Having an aircraft carrier means much better situational awareness and the ability to fly out and inspect things at 2,500km/h or so when needed...
Needless to say when the Israelis detected a US spy ship that was monitoring them break their agreements with the US so they pretended they thought it was an Egyptian ship and attacked it multiple times.... it wasn't the deck mounted 50 cal HMG that ended the Israeli attack... it was the call from a nearby US carrier group saying they had heard their distress call and Tomcats were on the way.... all of a sudden the Israelis realised it was a US ship and offered assistance.
When a US AEGIS class ship shot down an Iranian Airbus if the captain of the AEGIS class ship wasn't acting like a dick he could have called in air support from a nearby carrier, but the captains actions, including chasing Iranian navy boats into Iranian waters led the local carrier commander to call back his aircraft in case the AEGIS captain decided to shoot them down too.
They are not perfect.
In the Falklands war the tiny pocket carriers the UK had allowed the mission to be mounted but its lack of decent AWACS meant the carrier was kept back from the islands to protect them from Argentine attack meaning they were able to sink rather more ships than they would have if they had a real aircraft carrier there. The Ark Royal with proper AWACS and Phantoms would have been rather more effective and the losses of UK ships would be minimal or zero.
Cost cutting to save 50% on the price of a carrier led to the loss of half a dozen ships that cost way more.... not to mention the lack of a decent strike aircraft meant the mission to damage the runways on the islands made by Vulcan bombers could have been performed by deck based Buccaneers.