We call these the “stream grid” or “synthetic stream grid”, which are used mainly as a guide for delineation. If you click off of these lines, your delineation may only include a few grid cells near the point you clicked. The grid lines are not visible in version 3 until you have zoomed into a scale of at least 1:18,056, or in beta version 4 until you have reached zoom level 15.
The grids are derived from digital elevation models (DEMs) of the terrain, which have been processed to agree with the digital representations of the stream lines from either 1:100,000- or 1:24,000-scale USGS topographic maps. They aren't necessarily true streams, especially in headwaters areas. For most states, they represent 10x10-meter grid cells having at least 900 grid cells upslope from them, based on the DEMs used to do basin delineations. In states using a 30x30-meter grid cell, including California, Oregon, and Washington, they represent grid cells having at least 100 grid cells upslope from them. Outside of headwaters areas, the stream grids usually will agree fairly well with the streams that are shown on the USGS topographic base map in the user interface. When they disagree, it usually will be because either there is a difference in scale between the source data used to generate the grids and the source material used to create the base maps, or the source materials are of a different age and changes were made in the interim.
In more weathered terrain, with more humid climates, the stream grids tend to correspond to drainage channels, and will often coincide with streams. However, they are not intended to mean there is a stream in that place; only that if there were overland flow that these are the areas where that flow would tend to accumulate, given the terrain represented in the DEM.
It's not really possible to say, in general terms, that the synthetic stream grid shown in StreamStats represents the "natural stream path." What it represents is the path water would take given the digital representation of the terrain from the DEM. There really are too many variables to make any blanket statements regarding what the natural stream path would be. The DEMs generally, (but not always) give us a pretty good idea of the size of the drainage basin and other basin characteristics, and from those we can estimate the streamflow volumes we would expect in typical situations, using regression equations. The characteristics of a particular site, however, always could include some unique circumstances that could change things, and site-specific information should be preferred over StreamStats.