OSM Standard font size in QGIS composer

The OSM Standard style (CartoCSS) is designed for on-line viewing. Any printed version suffers from the problems you describe. In addition there are some other issues arising from it's nature as pre-rendered raster tiles:

• it's available at standard zoom levels. The layer is only likely to be readily readable when the scale you choose is very close to one of these zoom levels
• the text elements are baked into the raster tile so font styling etc is no longer controllable.

You will need to add additional layers in QGIS to achieve your goal.

Depending on your needs you could do one of the following:

• Add a number of OSM vector elements (the easiest ways is to use Overpass-Turbo and download geojson either directly or using a plugin within QGGIS) as QGIS layers, and style these in QGIS as an overlay. Often just place names help provide the relevant orientation information, but other features like the main road network may be necessary.
• Download all the OSM data for your area and build your base map using QGIS Styles.

In practice the former can lead gradually to the latter. There are probably already some decent resources which can help. It's always worth checking if Anita Graser (blog here) references anything.

PS. If I find anything I'll try & add it to this answer.