How do endothermic reactions absorb heat?
Because the system lowers its temperature, during the endothermic reaction the chemical system can absorb heat as a secondary process.
Because the system lowers its temperature, during the endothermic reaction. After that the chemical system (not the reaction) can absorb heat as a secondary process.
If the system is not thermally insulated, after the reaction some thermal energy will be transferred from the external environment into the cooled system, until the internal and external temperatures will become balanced again.
If the system in which the endothermic reaction occurred is thermally insulated, it will remain cold, and no heat will be absorbed at all (at least, not in short times).
The fall of temperature is caused by the endothermic reaction due to the withdrawal of kinetic energy from the system particles, that "slow down". This kinetic energy is used for the breaking of the stronger chemical bonds of the reactants that - in the endotheric reaction - is followed by the formation of weaker bonds in the produced substances.
In other words, as a consequence of the endothermic transformation, an amount of kinetic energy is transformed in an increase of potential energy, within the chemical system, without any change of the overall internal energy, if the system is isolated (energy is internally conserved). If the system is close, but not thermally insulated, there will be an income of heat, until the cooled system will reach the room temperature. The amount of heat that is absorbed in this step corresponds to an increase in internal enthalpy (or energy, if the mechanical work can be neglected or is zero because the volume is maintained constant in the transformation).