It is the apocrine sweat glands under the arms that emit the chemicals that indirectly cause body odor. I say indirectly because it is actually the microorganisms that live on the skin that need the chemicals from the sweat glands to flourish-and smell. The average human has a lot of micro-organisms (that is if you consider 1,000,000,000,000a lot) on their skin and hair. Usually these microorganisms are not only harmless but help keep away other, potentially more harmful, bacteria. Not everyone’s skin is able to support the same amount or types of bacteria in the same place.
Skin is acidic pH4-4.5 due to lactic acid in sweat as well as the acid that is produced by bacteria. One particular bacteria (Staphylococcus epidermidi) lives and flourishes in acidic conditions under the arm. It turns some of the sweat into isovaleric acid. This compound is the characteristic smell of body odor.
While poor hygiene can encourage body odor, some people and enthinicties (East Asians) have fewer apocrine sweat glands and therefore, have less issues with body odor (my friend Dorlene who is of Taiwanese descent is on such lucky person).
In alkaline conditions,(like that created by baking soda) it is believed that bacteria ceases to be attached to the skin and is more readily shed.