How would offering to help us affect that?
by showing that Russia has the capacity (allegedly) to help Western countries in times of crises rather than just relyingon them when something goes wrong in Russia. Putin's offer was largely made for domestic consumption -- to show its credulous people that America needs Russia and can't deal with a disaster by itself. That would then "lessen Russia's embarrassment" at having to have the Brits and the Americans pull the minisub from the ocean floor.