When people study apoptosis in cell culture models, a common method of inducing apoptosis is by simply withdrawing growth factors from the medium (usually called serum withdrawl). Just depriving the cells of growth factors causes them to shrivel up and die. What this suggests (broadly oversimplifying, of course), is that most mammalian cells require a constant input of survival signals from growth factors to avoid spiraling down into apoptosis. Moreover, if you subject cells to an insult, such as staurosporine or radical stress, that would usually cause apoptosis, you can often rescue them by giving them a slug of growth factors. Growth factors turn on cellular survival signaling systems that counteract programmed cell death.
How? Well, we could write a whole syllabus on that. But we're not gonna. We're going to focus on the broad brusthsrokes. Growth factors work by binding to extracellular receptors and triggering intracellular signaling events, most often mediated through the activation or deactivation of kinases (which phosphorylate things) and phosphatases (which dephosphorylate things). And among the many other processes regulated by growth factors via these "phosphate switches" is the process of survival signaling, mediated by diverse effects, from the inhibtion of caspases to the regulation of transcription to influencing Bcl-2 family interactions. As usual, a picture is worth a thousand words, and the figure and its caption are all you need to know for now.
Figure: Growth factor signaling. Very generalized cartoon of how growth factors run your life. Binding of ligand causes dimerization of the receptor, with consequent autophorphorylation on the cytoplasmic side. Thsi forms phosphotyrosine residues that serve as docking sites for SH2-containing proteins, which in turn signal to diverse cellular responses--including growth and survival, even in the face of a pro-apoptotic insult.