What happened?
Recently, the President of Iran Hassan Rouhani made a statement saying the rising tensions with the US calls for closer ties with Russia. He was speaking at the Shanghai Corporation Organization summit held in Bishkek on 14 June. This is significant as it makes it clear that Iran is looking out for any and every help it can get against the pressure it is under from the West led by the US.
Whether Russia is looking forward to such a relationship is unclear yet.

What is the background?

Iran has been under immense pressure from Washington following Trump pulling the US out of the 2015 Iran Nuclear Deal, citing it was 'the worst deal ever'. Iran eventually threatened it would itself pull out of the JCPOA and begin limitless enrichment of uranium. Following this, the tensions rose in the middle east, and the US deployed the USS Abraham Lincoln Strike Group citing possible threat to US assets and interests in the region. 

Any infringement would be met with ‘unrelenting force' was the American stance. Later Trump made a statement saying the US didn't want a regime change in Iran and said he did not share his NSA John Bolton's views.

What does it mean?

While the US has been extremely ambiguous in its policy towards Iran, the sanctions nonetheless have been adding pressure on Iran, putting it in a not so comfortable position. Looking for an ally in Russia seems to be its best bet right now.

Having been a victim of US sanctions since its Crimean annexation, Russia has been making its moves for increasing its global footprint through various agreements with countries like Turkey, China, North Korea whom all are facing brunt from the US. 
There is a growing anti-US sentiment in the air around the world and Russia has been riding the wave, taking advantage of every opportunity it could to oppose the global superpower, but Putin in May had said that Russia is not a ‘fire brigade' which ‘rescues everything' regarding the US pull-out from JCPOA. 

An intervention into Iran, while appealing for Iran may not be pragmatic from Moscow as it might further escalate tensions in the region that is already under threat of instability with four failed states and three ongoing wars where Iran is involved indirectly.