After every 210,000 blocks of Bitcoin that gets mined (roughly every four years), the block reward given to Bitcoin miners for processing the transactions on the blockchain is cut in half. This cuts the rate at which new Bitcoin is released into circulation by half. This is to create a synthetic form of inflation, that starts off high but then halves every four years until all Bitcoin is released and is in circulation.
This system will take many years however, and will continue until around the year 2140. After that point, miners will be paid for processing transactions and maintaining the blockchain with the transactions fees that network users will pay. These fees ensure that miners and what ever other systems that will be in place, will still have the incentive to mine and keep the network going and the blockchain stable. The general idea is that the competition for these fees will regulate themselves, only being profitable when there are fewer miners.
The Bitcoin halving is significant. It affects the price of Bitcoin because it marks another drop in Bitcoin's dwindling supply. The total maximum supply of Bitcoin is 21 million. This can never increase or decrease. As of mid 2021, there were about 18,715,050 million Bitcoins already circulating, leaving only 2,284,950 million left to be released via mining rewards into circulation.
Mining rewards are constantly dropping due to this halving. In 2009 when Bitcoin was first mined, the reward for mining each block in the chain was 50 Bitcoins. After the first ever halfing it was reduced to 25, then to 12.5, and then it became 6.25 Bitcoins per block as of May 11th, 2020.
To put this in another context and to use a similar real world example, imagine if this had to happen to gold. Imagine if the amount of gold mined from the earth was cut in half every four years. Gold's value is largely based on its scarcity, and so having an output of gold that halves every four years would theoretically drive its price up much higher.