There is a general aspect to this, and one related to physics.

Generally, it turns out that whatever one does in homological algebra when one wants to regard chain complexes as being equivalent if there is a quasi-isomorphism between them, then one is really working "in the derived category". Lecture notes on homological algebra that try to gently bring out this modern perspective includes these here and those pointed to in the references-section there. Notice that regarding chain complexes up to quasi-isomorphism means essentially to regard their cohomology groups as their "intrinsic information". This should be plausible to anyone who has looked into BRST and BV-BRST theory. There chain complexes of fields and gauge transformations appear and it is their cohomology groups which encode their intrinsic information (gauge invariant observables, anomalies, etc).

Now specifically on those derived categories that appear as categories of branes for the topological string in homological mirror symmetry. The modern mathematical way to understand the topological string is as an homotopy-theoretic TQFT in the sense of the cobordism hypothesis and the resulting concept of "extended TQFT" (which physically means: "fully local TQFT"). Here (triangulated) derived categories (and their genuine incarnation as (stable) \(\infty\)-categories) appear in reflection of what one might call the "higher covariant gauge principle" of physics, that cobordisms in TFT matter up to diffeomorphism, which matter up to diffeomorphism of diffeomorphism, and so on. Applied to the topological string (A-model, B-model etc.) this yields the mathematical formulation of cohomological topological 2d topological field theory (Witten 91) that was originally called "TCFT" (which is unfortunately a bit of a misnomer) (Getzler 92) following suggestions of Maxim Kontsevich, and the cobordism hypothesis-theorem for this case then is (in hindsight) the result of (Costello 04) which proves that and how these 2d cohomological TFTs are classified by their derived categories of branes. Derived/higher categories are absolutely essential for this, because while there are also more simple minded 2d TQFTs with values just in the plain category of vector spaces (and famously classified by suitable Frobenius algebras) the cohomological TQFTs appearing in topological string theory are not of this restricted form, they are richer.

Quite generally, from the modern perspective everything derived and higher categorical is the natural mathematical home for any concept at all (a sweeping-sounding statement that is however rather well supported, notably by the existence of foundations in HoTT and, I would add in the present context, of cohesive HoTT) and everything non-derived and lower-categorical is but some shadow which may or may not be available in special cases. Ultimately therefore the question is really the other way round: faced with some physics which is not described by derived/higher categories, one should ask which special simplifying assumptions do allow this. But of course there is still some way to go for that perspective to supercede the traditional familarity with underived/lower mathematics.