The films in the Marvel Cinematic Universe happen pretty much when they are released, with a few notable exceptions;
- Captain America: The First Avenger is set during the Second World War (but the date he was unfrozen in the present matches roughly the time the film was released - mid 2011).
- Despite being released over the course of three years, The Incredible Hulk, Iron Man 2 and Thor all happen in the same week, which is known as "Fury's Big Week" and is set at some point in 2010.
- Iron Man 3 was set at Christmas 2012 but was released in Spring 2013
This means, for example, that the events of Captain America: The Winter Soldier (released in Spring 2014) happened roughly two years after the events of Avengers Assemble (released spring 2012). Being as Captain America: Civil War has been released roughly a year after Avengers: Age of Ultron, we can assume that the two films are set roughly a year apart, meaning Captain America: Civil War is set about Spring 2016.
Futhermore, in Captain America: Civil War The Vision at one point states;
"In the eight years since Mr. Stark announced himself as Iron Man...
The original Iron Man was released in Spring 2008, further proving that Captain America: Civil War takes place roughly now.
Which means that when Peter Parker says he got his powers 6 months ago, he got his powers at some point in late Autumn 2015, post-Ant-Man and Avengers: Age of Ultron but (obviously) pre-Captain America: Civil War.
The nearest event in the timeline to the period where Peter Parker gained his powers would be the Agents of SHIELD season three episodes "Many Heads, One Tale" and "Closure", or some point during the middle of the season season of Daredevil (which seems to span from late-summer 2015 to mid-Winter 2015).
It's also worth noting that at the end of Ant-Man, Sam Wilson is looking for Scott Lang after their brief fight at the Avengers Facility, and a woman he's talking to tells him that they've got "A guy that jumps. A guy that swings. A guy that crawls up walls". As pointed out by @ibid in the comments on this answer, if we assume there is a time jump between the end of the main story of Ant-Man and this scene, the person the woman is referring to could be Spider-Man.