How can a reaction absorb energy to start but still be exothermic?

Exothermic reactions release their energy (heat) which is the reason why they make the surroundings feel warm. That's the reason why they feel warm, the energy that escaped during the reaction affects the surroundings.

Endothermic reactions absorb surrounding heat during the reaction, and the surroundings feel cooler because there is less energy in the form of heat surrounding.