I'm trying to explain what something means. Is it correct to express that using the word "idea" as follows?
- He pulls up next to him. To pull up (pause). That's the idea that someone does X.
- He pulls up next to him. To pull up (pause). It has the idea of someone does X.
An expression neither is nor has an idea: it expresses or conveys an idea. But more simply, it means.
So
That conveys the idea that somebody does X
or more usually
That means that somebody does X
Also, the idea of requires a noun phrase, not a verbal phrase, so
That conveys the idea of somebody('s) doing X
would be OK. (I put the 's in parentheses because some grammarians insist on it, but many people don't use it even in writing: see this explanation)
Your statement (quoted) doesn't provide a clear picture of your suggestion to someone. You may find the usage in any dictionary. For instance, on M-W:
Usage: It is usually considered correct to say that someone has the idea of doing something, rather than the idea to do it: he had the idea of taking (not the idea to take ) a short holiday
In your two examples, "It's the idea that ..." and "It has the idea of ..." do not convey any sensible or clear idea to me. Perhaps the phrase you want instead is "The idea is ...".
For example, after presenting an instance or two that in some way illustrate X, one says "The idea is that X occurred", to make explicit what the examples were intended to show.
