Here’s the short version:
Since the dissolution of the USSR, Russia had a large military / naval base in Crimea, by treaty with Ukraine. Russia’s lease on that base was set to expire in 2017. But, in 2010, Ukraine’s government made a deal with Russia to extend it until 2042. This angered Ukraine’s opposition and also the US. Ukraine’s opposition vowed to tear-up the extension deal if they formed government.
In 2014, the US overthrew Ukraine’s government, aiming to make Ukraine an outpost for NATO or NATO’s interests, and to kick Russia out of Crimea, which is an exceedingly valuable strategic point on the Black Sea.
In response to the US-orchestrated coup, Crimea, which had tried to leave Ukraine and rejoin Russia for decades, lawfully chose to join Russia. And the heavily Russian-ethnic regions of Donetsk and Lugansk in eastern Ukraine declared their independences from Ukraine and also sought to join Russia (but Russia didn’t accept them or recognise their declarations of independence at that time).
This infuriated the US and NATO, which then proceeded to funnel weapons into Ukraine and train Ukrainian troops and also Nazis, and encourage Ukraine to not stop trying to take Donetsk and Lugansk back by force. Years ago, Ukraine also signed the Minsk accords, under which Ukraine agreed to grant Donetsk and Lugansk autonomy within Ukraine. But Ukraine never upheld the agreement and continued to try to retake Donetsk and Lugansk by force.
After 8 years of Donetsk and Lugansk being brutalised by Ukraine, and in the face of increasing ceasefire violations (most of which were by Ukraine, according to the OSCE), Russia decided it’d had enough and finally recognised the independence declarations of Donetsk and Lugansk, and agreed to protect them from Ukraine’s attacks.
As Russia moved forces into Donetsk and Lugansk a day or two ago, the shelling of those regions by Ukraine’s army stopped for a while. That was the first calm those regions had had in years. But then, for who knows what reason, Ukraine started shelling those regions again, in the night and day, despite Russia warning Ukraine that it wouldn’t tolerate any ceasefire violations. In response, Russia has launched missile strikes and some other operations against Ukraine, taking out Ukraine’s air defences and air force, as well as other military targets.
Also, Russia recognised Donetsk and Lugansk up to the borders which they had in 2014, when they declared their independences. But since 2014, Ukraine had captured over half of their original territories. Russia will predictably push Ukraine’s forces completely out of the original borders of Donetsk and Lugansk.