You don’t in general, Inspector does not make parts, only changes some parameters of existing ones. To make a new part you need to either use Parts Editor (which is only partially finished) or create a custom part. For creating custom parts these two tutorials apply to the current Fritzing versions (most of the others are for older versions pre parts editor.)
if you decide to make parts this Inkscape extension is also very valuable for making proper schematics
Making a new breadboard is fairly tricky, and parts editor can not do it. Your best bet is to post the web site of the breadboard you want and I will have a look at making one (I have made several for folks in the past which are all posted here in the forums.) Once the part exists it can be imported and used like any other part.
As well you can right click on a part in the mine parts bin and select export part which will write the part as a .fzpz file to the file system. File–>open->filename.fzpz will import the file to your mine parts bin ready for use. Sharing parts is generally done by posting them to the “parts submit” section of the forum, or by providing a pointer to your part on github or other external file storage site. Finding parts is usually via google in the form of a search for “fritzing part part-model-number” which will find any parts available on the net. Often the images section of the search is the easiest place to find a part. Parts can be added to core parts by making a pull request against fritzing-parts on github. Assuming your part passes quality control (and the above tutorials will help with that) it will be added to core parts.
The main requirement for parts editor is that the starting part have the same number of pins as the part you need. I don’t often use parts editor, but AFAIK it will not create new connectors so it needs to start from a part with the correct number of connectors. As noted in the tutorial I tend to start from a part that is close and then edit the files to create a part. This requires that you understand how a part is created and what the xml does, but that is pretty much required for successful part creation anyway.
Peter