It sounds like what you've done (reasonably enough) has provided you with raw Openstreetmap data, rather than the map pictures which you're familiar with looking at. What you have is a set of points and lines, each with data attached to them - so for example a line might represent a motorway, a river, or a boundary (etc).
This is a good thing - if what you want is the data.
From what you say it's not what you are looking for. You want the pre-drawn ("rendered") pictures ("tiles").
As suggested by AndreJ - the easiest way (by far) to see these in QGIS is to use a QGIS plugin called 'Openlayers'.
Getting this working is very easy - in QGIS go to the menu item 'Plugins'|'Manage and install plugins' and find the one called 'Openlayers'. The installation process should just be a matter of clicking the obvious buttons. Once it's installed (and you exit the plugin manager) look for the available maps under the menu item 'Web' (this changed recently - if you have an older version of QGIS you'll find the maps under the 'Plugins' menu.
Depending what you want to do with the maps you may find that you'll either need the data, or the pre-drawn map tiles. (The words used to describe the difference between these two things are often 'vector' and 'raster' - look these up if you need to).
If you want to see an ordinary map using the data (rather than the raster pre-drawn tiles) you'll need to tell QGIS how you want these to look. QGIS doesn't necessarily know that you want a river to be blue for example. This is what others are talking about when they talk about 'styling' the data. This can be a very big job - but other people may have created styles that you want to use - but they won't probably look anything like the pre-drawn map tiles that you are used to using.
As you'll be able to tell, my answer only scratches the surface of what is a very big subject. Hope it's useful.