The "field" is where things like controllers are installed. As opposed to working on equipment in a lab or in your office.
Field software would be software that the field engineer or technician could use to diagnose and monitor equipment in the field.
"In-field" data analysis refers to analysis that is capable of being done in the field. You do not have to collect data blindly and then send it to someone in a different area who has different software in order to analyse it.
I don't know what they mean by "stakeout." They may mean their products reduce the amount of time you have to spend in the field trying to resolve a problem.