These so called "Tickets" are used to choose out a car that meets whatever the ticket is meant to be. Like the 1000 Tickets chooses out any car that is possible to obtain in the game. The B-Day Tickets chooses out any car that corresponds to its specified date, which assumes is your birthday. And Rally, Muscle, etc. Tickets chooses out any car that is within its specified class, like you get a Lancia Delta Rally Car from a Rally Ticket. And XP Level Tickets chooses any car that is within its specified XP Level, like you get a FGT from a Level 24 Ticket. It's popular because you're able to obtain cars that is beyond the 1 million trading limit, and able to obtain expensive cars. And the amount you obtain is infinite.
You obtain the ticket by:
1000 Ticket = Driving 200km or winning a single event 100 times (does not determine to your total records)
B-Day Tickets = Given when your birthday arrives (not necessarily always have to be truthful)
Rally, Muscle, etc. Tickets = Given from the Special Events section
XP Level Tickets = Given by getting gold on all 9 races in one series in either A- or B-Spec, or golding certain Seasonals
You can also get these by trading. Or if you feel like to cheat, you can dupe them once you obtain them.