what is the meaning of apple care services ?

It means that iPhone is covered by Apple Care.

Did you buy Apple Care+ or a protection plan from your network provider or the reseller you bought it from?

If you bought on a contract from a service provider it possibly includes protection and Apple care for the price you pay monthly for it.

Or did you buy this device used? The previous owner may have bough Apple Care for it, and the plan is tied to the device, not the user.

Either way, it means your iPhone is covered by Apple Care+.

Not sure where you are seeing this or looking for your Limited Warranty.

You can go to Check Your Service and Support Coverage - Apple Support to see the coverage status of your device.

You can also verify your coverage and Apple Care options directly on the iPhone in Settings ➜ General ➜ About ➜ Apple Care & Warranty.

So far, everything that you are telling us indicates that you did not buy a "new" phone, although the seller might have claimed it was new.

If the phone was purchased from an authorized Apple reseller, you will find their name listed when you search for authorized dealers in your area:

Find Locations

The message is very clear. If your phone requires service, you can contact Apple Support using this link:

Support