Q: "Why did this Lead continue to stay in a Campaign?"
Let's say the Fresh Leads 0-15 Days campaign has a Auto-Release Call Limit setting of 6. If the Prospect was not called seven times, they will remain in the Queue until they are past the Age Limit setting and you have completed any Pending Re-Calls for that Prospect. Prospects are not kicked out of a Queue until they are Dispositioned (called). To be sure your team is using the Call Queue effectively, you want to be sure you are calling all Pending Re-Calls within each Campaign, every day.
You may want to look at the Report showing all Re-Calls grouped by Campaign. This is an out-of-the box improveit 360 Report.
You can Filter this Report by Scheduled Re-Call Date/Time to isolate only Re-Calls that are Past Due, meaning your Call Center has not had time to work those Re-Calls.
Here is a Filter suggestion from our Team: 
For the two examples below and the question of "Why are these leads still in the Campaign when they are more than 15 Days old?"
Both Prospects were called on Feb 3rd and were not called again from the Fresh Leads 0-15 Days Campaign until April 5th and April 12th. Ideally, these Prospects should have been called a maximum of seven times between Feb 3rd and roughly Feb 16th or Feb 17th. If your team was working all the Re-Calls in the Campaign, these Prospects would have been called, and on the 7th call they would have been released from the Campaign.
Again, an Age Limit setting of 6 (six) means on the 7th Call they are released from the Campaign.
If the Prospect's pending Re-Call is never Actioned (called), even if the Lead Taken On date is more than 15 days in the past, the Prospect will remain in the same Queue until the Pending Re-Call has been completed (Prospect called and a Disposition button is clicked). This is why it's important to keep an eye on all Pending ReCalls.
These Campaign Members both functioned as we would expect. The Prospect had a Pending Re-Call and remained in the Queue until Actioned (called) on April 12th, at which time the lead was past the Age limit and was released from the Campaign. 
This Prospect had a Pending Re-Call and remained in the Queue until Actioned (called) on April 5th, at which time the lead was past the Age Limit was was auto-released from the Campaign. 
To prevent Leads from getting "stuck" in a Fresh Leads Campaign, be sure you are calling through all Pending Re-Calls within the Campaign each day. This assures that only Leads Taken in the last 15 Days will remain in the Campaign. Keep in mind, the system counts all days (weekend days too), so Monday your Call Queue will likely have more Pending Re-Calls to work through.
If you find your callers are having difficulty getting through all the Pending Re-Calls in a Campaign each day, you might consider changing the Auto-Release Call Limit to 3 instead of 6. Once the Prospect is called three times, on the fourth call they would be released, OR... 
You could also adjust the frequency of your default Re-Call settings on the Campaign so that Prospects are called more frequently, and all 7 Calls are completed in quicker succession:
Reporting on and reviewing your Call Center's Pending Re-Calls, will allow you to see trends in calling patterns and spot problematic build-up of Leads in a specific Campaign. Adjusting Campaign Settings assures your Leads are not getting stuck in a specific Campaign, and will flow organically into your other Call Campaigns. Reporting on and evaluating Campaign and Telemarketer Performance is something your Call Center Manager will want to do frequently as a best practice.
See also: Standard improveit 360 Out of the Box Reports and Dashboards
Q: "Why did this lead get called from this Campaign instead of a different one?"
Your Campaign Criteria should never overlap. Overlapping criteria can allow leads to fall into an unintended Campaign.
In the Example below, this system has a Campaign intended to isolate Today and Yesterday's Leads, and a second Campaign intended to capture Leads from the Last 7 Days (Fresh Leads 01 to 7 Days).
The Call Center Manager expected this Prospect to be called from the Today and Yesterday's Campaign however, the Campaign Criteria for the Fresh 01 to 7 Days campaign indicates:
Lead Source 'Taken On' is "On or before" 1 Day in the past (meaning any Leads Taken On or Before Yesterday). This line of criteria allows Leads taken Yesterday to be pulled into this Campaign.
In order for Campaign Criteria to be mutually exclusive you must insure there is no overlapping Criteria between any of your Call Campaigns. In this example, we will change the Criteria to:
Lead Source 'Taken On' is "Before" 1 Day in the past (meaning any Leads Taken Before Yesterday). This line of criteria would prevent Leads taken Yesterday from being pulled into this Campaign. Assuring that Yesterday and Today's Leads are pulled into the "Yesterday and Today" Campaign.
Q: Why doesn't this Prospect show up in any of our Campaigns?
- Has the Prospect been marked as Not Qualified?
- Does the Prospect have a past Appointment that hasn't been Resulted?
- Does the Prospect have a past Sale that hasn't been marked Final?
These are the first three reasons a Prospect would not Qualify for ANY Campaigns.
Check out our video series explaining the importance of Campaign Filter Criteria and mutual exclusivity