DNA is best known from the perspective of genetics, but its mechanical properties are also an interesting and important field of study. These mechanical properties play an important role in cellular processes such as replication and transcription. To characterize the mechanics of DNA, physicists measure force-extension curves of individual double-stranded DNA molecules. They observe that the DNA molecule cooperatively overstretches to a length 1.7 times longer than B-DNA at a well-defined force of about 65 pN: the overstretching transition. To visualize these DNA mechanics, fluorescent molecules are bound to the DNA via the process of intercalation. However, these intercalators are known to perturb the DNA structure and thus change the features of the force-extension curve. In particular, the overstretching transition is found experimentally to shift to higher forces than 65 pN, as a function of intercalator concentration. In this work, we develop multi-state freely jointed chain models to gain an understanding of the physical principles behind the effect of intercalative particle binding on the overstretching transition of double-stranded DNA. We show that a freely jointed chain like model with three possible segment lengths reproduces experimental force-extension curves, and that this model captures the physical principles behind the effect of intercalation on the force-extension curve. The three segment lengths represent B-DNA, overstretched DNA, and intercalated DNA. Moreover, our model agrees quantitatively with the experimentally found linear dependence of the overstretching force on the intercalator concentration. Finally, our theory predicts a further elongation to twice the length of B-DNA, induced by intercalative binding at every base pair, in the force-regime beyond the overstretching transition.