There are a few things at work here, that seem to be combined in your question.
The first is the concept that your child should be exposed to diseases at a younger age in order to avoid sickness at an older age. Much of your immune system works such that once you are exposed to a particular antigen (such as a protein from a virus or a bacteria), you produce T-cells and other related cells that protect against that antigen, and some of them persist for a long time - decades or even your entire life. This is how and why immunization works; it works the same way by simply getting sick. Here's a somewhat detailed explanation of memory in the immune system.
Chicken pox is the classic example of this from my generation; you knew you would get sick once from it, so your parents often tried to get you sick with chicken pox at 3 or 4, so you wouldn't have it at 7 or 8 - or even older, as an adult when it can be potentially more dangerous.
What is happening here is that everyone has a set of diseases that is in the local environment; this includes bacteria in your water, parasites, viruses, and bacteria in the dirt, and the various respiratory and skin-carried bacteria, viruses, etc., that the people in your community carry. You get exposed to these diseases as a child, have a reaction once or a couple of times, and then have sufficient immunity to be protected. Having your children in daycare means they get these reactions early in childhood rather than later. Other diseases - measles, the flu, chickenpox, etc. - tend to not work quite the same way (in the sense that you're not likely to be exposed to them just by going to daycare, though you certainly could be; and some of them, like the flu, are less easy to gain effective immunity from).
There is some evidence of this; this LiveScience article discusses a recent study that confirms that kids get sick the same overall, but get it earlier, when in daycare. This is intuitively logical; for diseases that you can be effectively protected from, you will tend to get them at one time or the other, with a comparatively similar cumulative amount. Diseases like influenza, which are harder to gain total immunity from, may not follow this pattern, but apparently either are insignificant in the total cumulative sample or you do gain some protection from (perhaps reduction in duration).
The second concept is the Hygeine Hypothesis, which refers to the concept that being "too clean" leads to more susceptibility to allergies and allergic diseases later in life. There is less support for this being related to common childhood diseases (although that support may be hard to obtain in any event, given the prevalence of immunizations); it's also thought to be most important very early in life (as a fetus, or in the first few months of life), and many things point to parasites as being more useful than viruses or bacteria here.
There is fairly significant support for this hypothesis, although it is certainly far from proven, both in its mechanism and the overall causal relationship. The hypothesis is only a few decades old (Wikipedia dates it to 1989, which my wife (an immunologist) thought was about right), so it still has some time to go before it's widely accepted as theory (both from needing more data, and more time for refutation).
The study you referred to in the OP may be either the study referenced in my first section, or the hygiene hypothesis - or it could be both, as it's entirely possible they're one and the same mechanism. The earlier study does mention asthma specifically, but it references it as if it were not part of the study ("Other studies show...").
I did find a study on The effect of cats and daycare exposure to asthma, although on a specific population of children deemed likely to get asthma (based on their having atopic dermatitis, a strong asthma risk factor). It showed a significant risk reduction for asthma for children in daycare; as with any study it's hard to say if other correlated factors could be relevant (and if the subpopulation is representative of the greater population).
The specific effects on asthma and allergies are likely to follow you through later in life, all the way through adulthood. The effects from starting daycare to avoid later sicknesses will likely be eliminated by mid-childhood (I've seen numbers around 10-13 years of age).
