(This is about U.S. treasury futures. Treasury futures in some other countries, like Germany or the U.K., are somewhat similar with subtle differences. Treasury futures in some other countries, like Australia or Korea are very different.)
The conversion factors are determined when the new futures contract is set up and don't change until the contract expires. The goal is to reduce (but not completely eliminate) the difference in coupon and accrued interest among the choices that can be delivered.
Imagine if all conversion factors were 1, and you had two eligible instruments, one paying 1% coupon, and another paying 2% coupon, same maturity. You don't need a very sophisticated cheapest to deliver model to see that the 1% coupon is always cheaper to deliver. Throwing in the conversion factor into the mix, so that more of the lower-coupon instrument needs to be delivered, levels the playing field between the choices, so that the cheapest to deliver may not be immediately obvious and may change with time.
According to CME https://www.cmegroup.com/trading/interest-rates/calculating-us-treasury-futures-conversion-factors.html
Every cash note or bond that is eligible for delivery into a Treasury
futures contract has a conversion factor that reflects its coupon and
remaining time to maturity as of a specific delivery month. A
conversion factor is the approximate decimal price at which $1 par of
a security would trade if it had a six percent yield-to-maturity.
A common misconception is that the DV01 of a Treasury security remains
fixed as the yield of the instrument changes. In truth, the
price-yield relationship of a Treasury security is nonlinear; as
yields fluctuate, the DV01 of a Treasury security changes.
If there was a U.S. treasury instrument with exactly 6% coupon, then the converson factor for this bond would be 1. But as of this writing, most coupons are much less, and so in order to get 6% yield, the price needs to be below par.
As an aside, according to https://www.risk.net/derivatives/7695186/cme-asks-clients-about-changing-implied-ust-futures-coupon , CME has been asking customers about possibly changing the UST futures implied coupon from 6% to 4%. This would result in higher conversion factors.