The Unibody MacBook released in Late 2008 is one of the last Mac Notebooks to feature a "user replaceable battery".
Apple removed this feature in mid 2009 when they discontinued the Late 2008 Aluminium MacBook and introduced the 13" MacBook Pro with it's larger 58 Wh battery and its higher 1000 cycle rating for the battery.
Continue by removing the plastic cover on the other side of the battery.
Insert a plastic opening pick in-between the aluminium casing and the plastic cover and pry the plastic cover out of the battery by bending the opening pick upwards.
With the plastic covers removed it's now time to get the battery pack out of its casing. However there is still one plastic piece glued to the aluminum casing which we have to remove first.
Insert a metal spudger in-between the plastic piece and the aluminium casing on the left side.
Be careful when working with metal objects near a battery pack. Shorting out the battery terminals can lead to a fire.
The plastic bracket may drop out of the battery pack when removing the pack from the casing.
When reassembling the pack make sure the metal pins on the plastic bracket are oriented in the direction shown as the plastic cover will hook onto these pins.