You can think of a tensor as "an element of a tensor product" (just like a vector is "an element of a vector space") and look at the definition of a tensor product via its universal property, it basically turns bilinear maps into linear maps. This universal property is elegant and immediately gives the Hom-tensor adjunction, but to show that tensor products satisfying it actually exist you'll need an explicit construction, which can be done by quotienting a free module. This explicit construction might look unintuitive at first, but you're really quotienting by the minimal relations required for the "bilinearity" property of tensors to work out