I have a bunch of polygons which are stored in a database. I would like to add them to the map in such a way that they can be edited using the leaflet-draw toolbar. Although
You have to add your polygons to the featureGroup drawnItems ! Let's say,
    var polyLayers = dbArray;
is your database array with polygons. First create a feature group with your drawn items:
    var drawnItems = new L.FeatureGroup();
and add it to the map:
    map.addLayer(drawnItems);
Then you simply need to iterate over your polygons from your database and add them to the drawnItems FeatureGroup:
    for(layer of polyLayers) {
        drawnItems.addLayer(layer); 
    };
Now the layers are added to the map and editable.
Here goes an EXAMPLE:
    var drawnItems = new L.FeatureGroup();
    map.addLayer(drawnItems);
    var polyLayers = [];
    var polygon1 = L.polygon([
        [51.509, -0.08],
        [51.503, -0.06],
        [51.51, -0.047]
    ]);
    polyLayers.push(polygon1)
    var polygon2 = L.polygon([
        [51.512642, -0.099993],
        [51.520387, -0.087633],
        [51.509116, -0.082483]
    ]);
    polyLayers.push(polygon2)
    // Add the layers to the drawnItems feature group 
    for(let layer of polyLayers) {
        drawnItems.addLayer(layer); 
    }