As a teacher at a preschool, there were definitely times when the kids did not want to go home and reasoning with a two year old about the matter often doesn't seem to help much. Although it should be great news that your kid doesn't want to go home (this is a result of the excellent care, fun staff and engaging environment your child has been exposed to after-all). It can also be a bit distressing for the parent, and create a challenging situation if your child is melting down into tantrums or fits of sobs at pick-up time.
Whether to avoid the issue altogether, or to help remedy one that has arisen, I recommend you offer your young child a "routine" to give him a sense of control and time to transition in any situation where he or she is expected to leave one location and move on to another. Repeatable routines allow kids to "wrap up" what they are doing both physically and mentally and help in switching contexts.
Specifically in relation to pick up time at preschool, you might try something along the lines of, "Hi (term of endearment and name) I'm here, you have about 10 minutes to get ready to go. Can you show me what you are playing?" Play with him for about five minutes and then say, "I'm going to talk to your teacher and gather your things. We have about five minutes until we leave." Talk to his teacher a little, gather his stuff . . . You get the idea. It's important to be consistent and accurate with the time estimates so that he can learn to transition.
It is important to give kids choices when you can. However, I recommend against bargaining in regard to more time. If you bargain now, you'll be bargaining every day and "Please, five more minutes" will become a common refrain heard even on days when you are in a huge hurry. If you'd like to offer your child a choice about something, I recommend something more like, "would you like to show me what you were playing for five minutes or just play the way you were for five minutes?"
When you tell her it is time to go, you might try something like, "Would you like to hug (teachers name) or blow her a kiss today?" and/or, "Would you like to put your lunch leftovers in your bag or would you rather I did it?" In doing this you establish that it is time to go, but you are also giving her an opportunity to choose how she goes about leaving.
Try to keep leaving fun, most of the time you can simply engage your little one in conversation. For example, you might ask what his favorite part of the day was. However, every once in awhile (and especially if you've been having trouble at pick up time) make a game out of getting to the car. How many giant steps does it take? How many baby steps? Assuming you'll need to cross a parking lot, I don't suggest actually racing to the car, but if traffic and safety aren't a worry, why not race? If it is fun, it will be more appealing to the little one.
If you've been having a challenge at pick-up time and that is why you are reading this article, don't expect change overnight. It will take a few pick-ups with a calm administration of the "routine" before your child starts to connect the dots. If your child is simply refusing to leave, please do not use the "alright, off I go then" bluff. If you are only bluffing and they call your bluff, you lose credibility which does not really help you in the end. Simply and calmly, scoop up your child with an empathetic, "I know you were having fun and would really like to stay, but it is time to go home. You will be back and play more again (Tomorrow, Next Monday. . . whatever is most appropriate here)." Carry your child to the car in as calm a state as you can muster and don't say anything more until your child has calmed.
While you wait for that day to come you can "notice" how the behavior at pick up time effects his play time. However it is important to "notice" this after the drama is over and not during the drama of a tantrum or fit of tears. It is also important that the "noticing" not be casual but not be disciplinary either. You might say, "Hey (term of endearment or name here), I know you really like school and I've noticed you are having a hard time when it is time to leave. Because I noticed that, I am trying to give you a five minute warning and let you play just a little bit after I get there. Today, instead of hearing me tell you that you had five more minutes to play, your were fussing so loudly you used your five minutes up having a fit." Or, you could try, "I really wished you had been willing to show me what you were playing. I had five minutes to play with you there and wanted to, what a bummer."
After using a routine and these methods for a little while, a good day will come. The first time it does work, when you get to the car, notice that she went with you without a bunch of fussing and crying about it. "I notice leaving day care today seemed easier for everyone. What nice choices you made."
Along the way, remember, you are having this problem because you chose a wonderful day care or preschool and be reassured you are making the right decisions in your child's care.