The answer by @discreet gives good links, but I want to add my understanding.
Without knowing which vendor you bought your PKI from, it's hard to know exactly what the PIN is doing inside, because unlike EMV bank cards, enterprise PKI systems have no industry standards and each PKI vendor will do things differently. So here's my response to your two points based on how these things usually work:
There is a certificate embedded in this card which contains my private key.
Correct. Usually your PKI smart card will have a mini-computer in in that powers up when you connect it to the card reader. For this reason, your smart card is more than a storage device (like an SD card), rather it is its own computer. Your PC can send requests to the card like
give me your public key,
sign this data,
decrypt this data, etc, the card will do the processing on-board and send the data back. The card can even refuse to do certain requests, such as
give me your private key.
One of the operations that all smart cards support is
generate a new keypair. The card will generate a new private key (which will never leave your card, if you specify that option), then the corresponding public key will be passes back to the PC to be made into a certificate by a CA, then handed back to the card as a certificate for storage.
My 6 digit PIN is the key to encrypt my private key
Mostly Correct. This is the part that varies by manufacturer. Even though the private key never leaves the card, you still don't want to store it in plaintext because a careful attacker can remove the flash memory chip and extract the data. My guess in that your 6 digit PIN is stretched into a symmetric encryption key which is used to encrypt / decrypt your private key while it is in storage on the card. This means that until you give it your PIN, even the card can't access your private key.
The PIN is never stored anywhere, except for in the card's RAM while it is plugged in. When the card powers up, it will ask for your PIN, if that allows it to properly decrypt your private key then the card can go ahead and do crypto things, if the PIN you gave does not produce the private key which matches the certificate (ie it produces garbage), then the card will throw an
Incorrect PIN Error.
A quick Google shows that many smart card vendors post on their website a detailed whitepaper on how their smart cards work. I would encourage you to figure out which vendor manufactures your smart cards, and continue your reading on their site.