I am interested in making short videos and films.  I am interested in
producing my own as well as others.  What does a production company
offer that I cannot just do myself?  However, if I am to start a
production company I have no idea how to even begin such a venture. 
How would I create my own production company?  I live in Calgary,
Alberta, Canada.

Making videos and short films is the easy bit.

Just buy some kit and start shooting.

However, the tricky bit is marketing what you've created, if that is
also your intention.

Who would want to see it?

More importantly, who would pay to see it?

Maybe for starters, you should shoot the Calgary Stampede?

Because, let's face it, there's nothing else in Calgary, is there?

(I've been and seen it all.)

You ask what a production company does that you can not do and then go
on to say you have no idea how to even begin.  That's where the
production company has you - they know what they are doing and can do
it well.  Why not intern or get a job at a production company and lean
the business first.

"Because, let's face it, there's nothing else in Calgary, is there?

(I've been and seen it all.)"

thats a bit severe isn't it?

If you make it on your own, you would not be having any patent and
copyrights on the film or video and its like an open source or open to
piracy but whereas
a production company offers patent rights authority & copyrights of
the short film or the video that you are going to make.
Hence, you can legally fight if someone pirates your film or video.

If you're the creator of the film, you own it, and the copyright belongs to you. 
But, if you want to make a show for a TV network, the network won't
just give you a contract and a check--you need to be able to finance
the entire production yourself, before they'll send you any money--and
offer proof of production insurance, E & O insurance, etc. That's why
networks, studios deal with production companies. Certainly you can
make a film yourself and then try to shop it around to someone who'll
distribute it--lots of people started this way.