How do you explain the usage of "with"?

I'm looking for a logical explanation of how we use the preposition "with" in the following context:

The company took the first place with their competition a distant second.

North America had the greatest number of travellers over the 5-year period with Central and Eastern Europe showing a similar pattern.

I suppose my main question is how do I explain the phrase (it's not a clause) that follows the "with"? and how do I explain the function of "with" in these sentences?

It's strange I can't find a duplicate and I believe there is one. I will try my best to explain it, but it is not easy.

The preposition with has a very important function as the below definition indicates:

used as a function word to indicate an attendant fact or circumstance: 'He stood there with his hat on'

The attendant circumstance means an accompanying circumstance. In other words, the action (or state) described after "with" is happening at the same time (usually).

The example in the block quote above could be rephrased to:

He stood there and he was wearing (on) his hat at the time (he was standing).

Your examples:

The company took the first place and their competition took a distant second (at the time the company took the first place).

North America had the greatest number of travelers over the 5 year period and Central and Eastern Europe showed a similar pattern (at the time North America had the greatest number of travelers...)

When you have something, you are with something. If two things happen at the same time, one thing is happening with the other.

Some more examples:

The matter was resolved with both countries cooperating.

She sat silently with the cat dozing at her feet.

The man is leaning against the wall with his arms folded.

You need to read as many examples as possible and get yourself familiarized with them.