How does a blackbody actually heat up and change temperature if it emits all the radiation that it absorbs by definition? And by extension, if the absorptivity and emissivity of a body are equal, then how is the temperature actually changing?
To begin with, in real life the absorptivity and emissivity are not exactly the same as both of these things are functions of wavelength. This means it is possible for an object sitting in sunlight to absorb well across a range of wavelengths but if it emits poorly in the infrared then it will eventually become hot enough to burn your hand if you touch it.
Furthermore, the whole blackbody derivation relies on the black body being at the same temperature as its surroundings: this represents a state of equilibrium in which the heat going in equals the heat going out and temperature therefore remains constant. While a cold body is being heated, it own blackbody emission spectrum does not match the blackbody emission spectrum of its environment and heat transfer is biased in the direction of the cold body getting hotter.
A blackbody is in thermodynamic equilibrium. It is not getting hotter or colder.
Having realised this, then it is more straightforward to understand why a true blackbody emits as much radiation as it absorbs.
If an object is being heated, in the sense that its temperature is increasing, its spectrum may still approximate to a blackbody if the timescale on which the temperature is increasing is much longer than the timescale for it to achieve a thermodynamic equilibrium - i.e. if the temperature increases slowly enough to model it as the evolution through a sequence of quasi-static temperatures.
it emits all the radiation that it absorbs by definition?
This is not the correct definition of a black body. The Wikipedia definition of a black body says:
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.
In other words a black body is defined as one that absorbs all incoming radiation. Not as one that emits all the radiation that it absorbs.
Indeed, any object at thermal equilibrium in purely radiative heat exchange will emit all of the radiation that it absorbs, whether it is a black body, gray body, or white body. Your definition is a definition about thermal equilibrium, not about the type of object that is in thermal equilibrium.
While a black body is colder than its environment it is absorbing all of the incoming radiation. It is emitting a characteristic black body spectrum, which contains less total power than the absorbed power. Thus it gains thermal energy until it reaches the equilibrium point and behaves as described above.