I haven't used the sensor myself and couldn't find any PICAXE code which has used it but it does look like it can be used with a PICAXE.
It is different to I2C Eeprom in that it uses 'commands' rather than 'register addresses' but I believe it should be usable with standard PICAXE HI2C commands, and, if not, then via I2C bit-banging.
It looks easy enough to extract the humidity and temperature from the returned data. Both are linear values so should be easy to convert to real world representations.
There are a couple of details I am not clear on from the datasheet 'papaof2' linked to; 7.4.1 initialisation and calibration, and 7.4.3 where it's not clear if the full data can be read while status shows busy.
That might mean some experimentation, trial and error, digging for clarification, but there does seem to be example code for other microcontrollers which may give enough of an insight for getting it done.