Why does most of the world's international trade take place between
economies that are similar in their level of economic development?

Here are a couple educated guesses:

1) Location: It is much cheaper and easier to trade with a neighboring
country rather than a country far away.  The largest trading partners
in the world are Canada and the US.  European countries tend to trade
amongst themselves.  Zimbabwe trades with S Africa, Ethiopia trades
with Djibouti, Argentina trades with Brazil.
And notice that countries in a region are very likely to be in a
similar level of economic development.

2) Desired goods: Highly developed countries have high demands for
much different goods than lowly developed countries.  The US wants
semiconductors, DVDs, plasma TVs and luxury cars... People in Nigeria
don't produce these things because there is no demand in Nigeria for
these products.

3) Understanding the market: If I (here in the US) want to sell goods
in Swaziland, what approach would I take?  I have no idea how to sell
my goods.  I don't know who to talk to, how to set up a shop, how to
market...  it is a whole different world.  However if I want to sell
my goods in Europe or Canada, I could figure these things out much
easier.

In the world market for agricultural products  - sugar, grains -  you
also have the problem that USA and the EU protect (actually subsidize)
their own producers, thereby limiting trade with third world
competitors.