Hemoglobins affinity for oxygen is different in the lungs vs the body’s tissues and the tissues (especially muscle) have a protein called myoglobin which is even better at binding oxygen than hemoglobin. In the capillaries of the body’s tissue the hemoglobin has lower afinity for the O2 so it falls off and can diffuse from the RBC to the cells that need it (think of it like hemoglobin pushing it off and myoglobin acting as a vacuum sucking it in).
[This curve](https://en.m.wikipedia.org/wiki/File:Oxyhaemoglobin_dissociation_curve.png) shows how hemoglobin’s affinity for oxygen changes. On the bottom is the partial pressure of oxygen, more O2 present means more will bind hemoglobin. But also pH, temperature and metabolites will affect hemoglobins binding. These all add up to conditions which strongly favor O2 to bind hemoglobin in the lungs, but then change so that it will dissociate from hemoglobin in the tissues (where there is lower O2 partial pressure, lower pH, higher temperature and more DPG).
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