It's a Lithium battery. Those connections don't go directly to the battery they go to a control IC which allows the battery to be charged and discharged safely. Personally I just rip those off and get the battery down to it's raw terminals. You do however have to be very careful with lithium batteries but if you observe a few basic precautions they're quiet safe. The biggest problem when you have a raw lithium cell is to never EVER accidentally dead short the battery terminal together. Never discharge the battery bellow about 2.7 volts, charging should be constant current at 1C until the batteries voltage reaches 4.1 volts, and then it should go into constant voltage (4.1 volts) until the current drops to 10% of it's 1C value, then the battery is fully charged. Never trickle charge a lithium battery, never over charge one, and never EVER dead short it's terminals or fire can result. If the battery ever gets warm during a charge cycle immediately stop as something is wrong. Lithiums have a charge efficiency of 99% so they should stay at room temperature during charging.
If you don't want to deal with the second stage of charing (constant voltage to 10% of the intial charge current) you can skip it, but the battery is only about 80% charged after the constant current stage.