dBm are absolute power, they correspond to a certain amount of mW. For instance 0dBm = 1mW, 33dBm = 2W.
dBi are relative power: they indicate how much gain is given to an input signal. It’s like the “dB”.
For instance :
0dBm + 3dB = 3dBm = 2mW
3dBm + 3dB = 6dBm = 4mW
6dBm + 3dB = 9dBm = 8mW
9dBm + 3dB = 12dBm = 16mW
As you see, 3dB is not equal to xx dBm. It is a factor which means the power (in watts) is multiplied by 2.
An antenna “multiplies” the input power by a certain factor. It’s “relative power”.