Some folks believe that vaccinations "overwhelm" a child's immune system and may make the child more susceptible to illness in general. For example, you will find lots of non-vaxers saying their non-vaxed kids don't get "sick" as much as the vaxed kids they know. But that's pretty anecdotal.
Essentially, this study set out to prove that vaxed kids don't get sick any more than non-vaxed kids. Their findings were that, just to give an example, being vaxed for polio wouldn't make a person more likely to become sick with pneumonia.

It sounds kind of dumb when you put it that way, but actually these are the kinds of studies that OUGHT to be done. A lot more of them!
Couple problems with getting too excited about this study: First of all, most people with vaccine concerns are worried about auto-immune diseases (MS, diabetes, allergies) or neurological damage (ADHD, autism) rather than infectious diseases. This study didn't address auto-immune diseases at all.
Second, the study has many variables and no real control group, from what I can tell. I'd like to know what they considered an "unvaccinated participant." Would that be a child with absolutely no vaxes at all? Or does it mean they compared a child with Hib-polio-DTaP vaxes to a child with only Hib-polio to try to come to some kind of conclusion about the DTaP? Do they actually have good enough centralized record-keeping in Sweden to do this kind of study with longitudinal data and get reliable results? The best studies match a patient with one variable to another patient who is similar in every way except that variable. That wasn't done in this study.
Third, there is a vast continuum of "sick" between, say, frequent minor colds versus pneumonia requiring hospitalization. The study found no association between vaccination status and being hospitalized for an infectious illness. I'd be a lot more interested in the results of a study that looked at association between vaccination status and frequency of minor viral illnesses, or vaccination status and being hospitalized for asthma.