Leading zeroes
The number you refer to is usually called the mining difficulty or target.
The aim of miners is to produce a block with a hash less than that target.
If you represent numbers in binary, a number with more leading zeroes is a smaller number.
For example 00010 (2) is smaller than 00101 (5)
However you shouldn't think about leading zeroes because 00101 (5) is smaller than 00110 (6) but bigger than 00100 (4) although all have the same number of leading zeroes.
It is an arithmetic relationship that is tested, not the number of leading zeroes.
Target value and mining
The target is adjusted up and down regularly. It doesn't continually increase.
The target is adjusted to make the average time to find a new block close to 10 minutes.
This means as there are more miners the target is likely to increase. But more importantly, if there were fewer miners but each had more powerful hardware, the target would also increase, not decrease. It is the total amount of mining power, not the number of miners, that affects the target.
In the past, when the value of bitcoin fell, the value of the mining reward also fell. This can result in less efficient miners leaving the mining business and the difficulty target dropping not rising.
Miners assemble a block of transactions selected from a pool, then calculate the hash, if too big, they make a small change to the block, like change the nonce value, and recalculate the hash, they repeat this until either they find an arrangement that hashes to a value smaller than the target or until some other miner beats them and they have to start all over again.