These electron carriers are important for the creation of energy, primarily. We will focus on two things: driving the reactions in glycolysis and the Kreb's Cycle and transporting electrons for the electron transport pathway.
`NADH` first comes up in glycolysis as an oxidizing agent (`NAD^+`) that oxidizes glyceraldehyde-3-phosphate and becomes reduced to `NADH`. It does the same in the Kreb's cycle to oxidize isocitrate, `alpha`-ketogluterate, and malate. `FAD^+`, similarly, oxidizes succinate in the Kreb's Cycle. Looking at these reactions, you can see that these two molecules are important to keep energy production happening! If any buildup of product occurs, then these processes stop, and energy production shuts down. Without energy production, the life of the cell would end!
Of course, the energy production seen in just glycolysis and the Kreb's cycle is not sufficient on its own to keep up a living eukaryotic cell! These two factors are also essential to keep up the primary energy production process in a eukaryotic cell: the electron transport chain. The reduced forms of these molecules, `NADH` and `FADH_2`, interact with proteins in the inner mitochondrial membrane to produce a hydrogen ion gradient. The gradient of hydrogen ions across the inner mitochondrial membrane allows hydrogen ions to flow through ATP Synthase, powering ATP production. ATP Synthase, being powered in this way, is able to produce (effectively) 2.5 ATP per `NADH` and 1.5 ATP per `FADH_2`.
Thus, you see two primary ways in which these electron carriers keep cells going! There are other ways, but these two are their most important and universal roles.