Knowing how and why mites use bees as a vehicle to get around in the hive makes it understandable that they are not easily seen. When first coming in to a hive, the mites typically arrive from the outside on the older age foraging bees that may pick them up from a flower or perhaps while Robbing another hive. Or bees coming that are immigrating from a "mite bomb", a hive dyeing from varroosis - death by varroa transmitted viruses. Surviving bees will look for other colonies to join. (See link below for more on mite immigration)
Once inside the hive, they then look to transfer to the younger age "nurse bees" so they can travel to the area of the nest where the nurse bees are feeding young. The heart of the nest is the prime target for the mite's "breeding grounds" and source of food. The mites have evolved to recognize these different age bees by there scent and take advantage of any opportunity to switch from one to the other. 
Studies have shown that the mite may spend as much as 20 minutes on a single bee while waiting for the opportunity to change bees or get off a bee and into the comb with a larvae. During this time, they are able to go undetected by masking their own scent to match that of the host bee they are traveling on at a given time. Similar to how the little green Anole lizard in the garden will change it's color to match it's surroundings when moving from a green to a brown surface to better camouflage into the background. 
Upon arrival into the heart of the nest, the foundress mite can then dismount the bee and invade an open cell area to search for a fifth instar larvae. The mite knows, by scent, this is the larval stage where the nurse bees will soon put a cap covering the cell for the 13-14 day pupation period. The mother mite can begin her egg laying process in this environment, hoping to go without being detected for about two weeks. 
The first egg the mite lays will be a male, then she continues laying more eggs every day or two, which will all be female. The male then begins mating with the females, so by the time the developing bee does emerge, on average there will be two to three mature pregnant daughters. These offspring must search for an adult bee to feed on so her hormones can fully develop and she can start the cycle over again.
So that's an average of 2.6 additional mites emerging per infected bee, so doing the math, it's easy to see how the mites numbers build so rapidly. The more mites there are per bee, the higher the virus load transmission can occur and start to have negative effects on the colony.
Having the knowledge of how mites exist without detection and ejection by the colony, helps to know when and how we will best go about Checking the Bees and Treating for Mites.